This relies on perfect luck for more than just the stacked deck, but it otherwise fits the bill perfectly.
1: "Draw" and play Karakas, exile Simian Spirit Guide and Shinen of Life's Roar to waste the Kira protection on Elesh Norn, then tap Karakas to bounce it. 5 cards in hand.
2: "Draw" and play City of Traitors, tap it, and exile another Spirit Guide to play Daru Sanctifier face down. 3 cards left.
Pitch a spirit guide and tap Karakas to turn Sanctifier face up. Choose to destroy the Nevermore that's naming Mana Clash. 2 cards left.
Pitch a fourth spirit guide to play Mana Clash. Flip 21 straight heads, against 20 tails and a heads.
Given the privilege of just one face up spell, Aura Barbs wins easily against this opponent without needing to manipulate 42 coin flips, but it also needs more mana.
"Hate bears" typically refers to low-mana-cost creatures that have an ability to hose certain types of cards, such as Gaddock Teeg, Kataki, War's Wage, or Aven Mindcensor. In its truest derivation, the term "bear" refers to a 2/2 creature for 2 mana, but the toughness is usually discounted as a less important trait so that 2/1s still qualify, and as the Mindcensor shows, some people will still consider them "bears" even if they don't cost 2.
Like you might expect from the card text, Blood Moon is heavily used in Modern against decks with lots of nonbasic lands (and in decks that either run a sizable contingent of basics or don't mind their mana being red anyway), while Ensnaring Bridge is a catch-all, long-lasting answer to almost any creatures, particularly in decks that can play out a hand full of low-cost cards or discard cards to keep the bar set at 0.
In Betrayers of Kamigawa, one of the theme decks was Rat's Nest, which included a copy of Umezawa's Jitte. The natural response to an outlier like that was obvious: that particular deck got bought up everywhere, leaving the other three decks neglected on the shelf, and some venues (the ones who are actually familiar with the prices of cards) decided to raise the price on that deck so that they'd get more money (and people still bought it at the higher price, especially once all the other places that didn't raise the price were sold out, so it worked out well for those stores).
Curse of Echoes is optional for the opponent. Unless you're having them cast the original spell somehow, Curse of Echoes is simply a blank, as they'll choose never to create their own copy if creating one would benefit you.
Fork effects are not as useful as additional Rites and other variants like Twinflame. With a Fork, all you do is create a single copy of the spell, and that copy was not "cast" so it never picks up any Precursor triggers. A second copy of the token spell does more than all of the Forks put together, with the exception of getting Eye triggers, a purpose for which they can defer to any blank instant or sorcery: 2 Rite + 39 blanks is better than 1 Rite + 40 Forks. 40 token spells, of course, would be better still, if you're just going for a brute force "do one thing over and over, one card at a time" routine.
Kicking the Rite is also not really a big deal--in fact, given that you have at least 3 copies of Doubling Season (which you will), and at least one token copy of everything important produced by some other means, a mere Sundering Growth with its two-mana cost (that you don't actually have to pay, of course) works out to produce more tokens than paying 5 or 9 mana for a Rite. Yes, even though you are destroying all of your artifacts and enchantments, over and over again. The trick is that Populate is a non-targeted instruction that allows you to produce a token of your most valuable permanent, even on the copies that are nominally "targeting" something else. Here, almost all of the populate steps should go after Doubling Season; you only need to make tokens of anything else once each per round of Precursor Golem spell copies, and only just in time to protect their last surviving specimens from being destroyed by making more tokens that will persist into the next round of copies.
Eye of the Storm and Precursor Golem are also not as much of a combo as would be ideal. You only get a number of Precursor triggers equal to the number of Precursors that exist at the moment the spell becomes cast, and eye of the Storm has you casting multiple spells in succession in the middle of the trigger's resolution, where there is no way to amp up the number of tokens in between casts. You can get a more effective way of scaling by doing away with Eye of the Storm entirely, thereby letting your spells go to the graveyard, then using a sequence of spells that can get each other back from the graveyard (but only in one direction). For example, you could use Sundering Growth to make your tokens, then get that back with Reborn Hope, get Reborn Hope back with Revive, and get Revive back with Mystic Retrieval.
In addition, to get any benefit of scaling at all, you'll want some way of dynamically creating copies of those spells. They don't target permanents, so Precursor Golem doesn't contribute anything at this level, but one neat thing you can do is add an animated Pyromancer Ascension, which you will of course be making token copies of, or else Mirror Gallery + Mirari. In the latter case, you need a source for repeatable generation of mana, since it costs 3 mana per copy of the spell. In the former case, you need ways of "powering up" your Ascensions, because they don't do anything without counters on it, but clearly with Doubling Season you only need 1 trigger per copy to leave it with a lot more than 2 counters. There are a couple different ways of doing this: you could play Mystic Speculation once without buyback, then play another copy with buyback, and for 2 mana you can immediately power up every copy of Ascension that currently exists on the board. There are also a few spells that can return to your hand for free, allowing you to avoid mana altogether: Petals of Insight (although this can potentially interfere with your deck setup, allowing you to go infinite if you have a way to put cards back in the library), Molten Birth, View from Above, and potentially even Titan's Revenge for 0 if you can lose the first clash (or get the initial copy of the spell countered for an illegal target) but then win all the rest. A caveat here is that if you do use spells like Reborn Hope, you can't have any cards that are capable of countering them (such as Odds//Ends), or else you can put it in the graveyard in time for a copy of the spell to return its original self, quickly going infinite.
There's clearly some space for having a few nested operations among instants and sorceries, but the trouble is it's rather limited, and leaves you with a ton of superfluous deck space that isn't appreciably helping. Being able to use that deck space on permanents that have more room for granularity than the broad "Sundering Growth, make lots of copies of everything" will just be more powerful.
1: "Draw" and play Tolaria
2: "Draw" and play Karakas, tap Tolaria targeting Elesh Norn and triggering Kira (oh no, we don't get to make it lose bands with other!), then tap Karakas to bounce Elesh Norn
3: "Draw" and play City of Traitors, tap it for 2, exile a spirit guide, and play Daru Sanctifier face down (which survives, now that there's no Elesh Norn). 5 cards in hand.
Exile a second spirit guide and tap Karakas to turn Daru Sanctifier face up. Choose to destroy the Nevermore that's naming Echoing Truth. 4 cards in hand.
Exile a third spirit guide and tap Tolaria to play Echoing Truth on a copy of Nevermore; all of them will be bounced. 2 cards left in hand.
Play Black Lotus followed by Timetwister, and the rest is easy. The card draws will be replaced by Uba Mask, but you still have access to them for the turn, and nothing's stopping you from playing them.
If you're building for an actual, competitive decklist, remember that deck space economy is a thing: you certainly don't want to run a combo that takes up half the slots in your deck. Through the Breach and Footsteps of the Goryo are the two main ways of getting a temporary creature followed by a death trigger, and the most efficient Protean Hulk package weighs in at just 4 cards.
With the first trigger, you grab Viscera Seer and Body Double, where Body Double copies Protean Hulk. Sacrifice the Body Double and you get to fetch another set of creatures, this time Mogg Fanatic (or Death Cultist) and Reveillark. Sacrifice Mogg Fanatic to its own ability for 1 to the dome, then sacrifice Reveillark to Viscera Seer just to get a trigger. Now you get to reanimate two creature cards with power 2 or less; choose Mogg Fanatic and Body Double, and this time have the Body Double copy Reveillark. From this point, you can keep sacrificing and reanimating both of those creatures forever, or at least until the opponent loses.
Obviously, since this loop can be repeated without limit, it isn't eligible for consideration in the task at hand. But it does go to show the power of recursion: in this case, being able to use Protean Hulk's ability twice means you can get a more potent set of creatures than if you only used it once. Just plonking down a huge swath of creatures and attacking for 100 or so is nowhere near as effective as building a deck around repeating some operation many times. The growth rate of each repetition is also a factor: in the naive 104-damage package I listed, each additional creature only provides a linear contribution of 4 damage. Meanwhile, if Hulk allowed you to fetch 7 mana worth of creatures rather than 6, you would be able to get Hellraiser Goblin (the cheapest fetchable source of haste) alongside Goldnight Commander, which causes each creature to boost the entire army--a quadratic growth rate, such that you could attack for 882, or even more if you decided to add Shield Sphere or the various X-mana creatures that "come into play dead". Even quadratic growth can be made to look puny within the confines of Magic: the most-cited example of a fast-growing mechanism is Precursor Golem + multiple copies of Rite of Replication, although in some cases a two-mana Sundering Growth is capable of exploding even faster than a nine-mana Rite, and there are mechanisms that can get even faster still. You just have to know where to look, and a game with over 15,000 pieces provides an awful lot of possibilities to look through.
There are much better things to be Flashing in than Protean Hulk. For instance, Academy Rector, which can bring out Omniscience. Flash isn't banned in Vintage, only restricted, but there are still much better things to do with your time.
That Disciple of the Vault package is far from optimal for a number of reasons: it never causes any "damage" at all, only life loss; the life loss is in increments of 1 at a time so the game will end abruptly while 28 triggers remain unresolved on the stack; and you've only used 4 of the allotted 6 mana. For instance, you could focus on the 0-mana creatures that do stay alive: 4 each of Ornithopter, Phyrexian Walker, Memnite, Dryad Arbor, Crimson Kobolds, Crookshank Kobolds, and Kobolds of Kher Keep (28 creatures--you could also add Shield Sphere but it wouldn't do any good here since it has defender). To go with that, add Hellraiser Goblin (3) and Mirror Entity (3). Tap all four Dryad Arbors for mana since they now have haste, pay for Mirror Entity's ability at X=4, and swing with the 26 untapped 4/4 creatures for 104 damage. And still these two- and three-digits numbers don't get you very far in the grand scheme of things.
That is infinite. Praetor's Counsel, respond with Increasing Vengeance for free, respond with Battle Hymn for free. Get some mana, make a copy of the spell, and now Battle Hymn and Increasing Vengeance are in the graveyard. The copy of Praetor's Counsel will return both cards to your hand (and exile itself, which is pretty meaningless because it's a copy). Then you can replay both again, for as much red mana as you want.
If you have a source that will get you, say, 100 green mana but no more, it's best to get all of the mana at once. Then the layers stack on top of each other as (green stage with 99 mana floating) under (green stage with 98 mana floating) under (green stage with 97 mana floating) ... under (green stage with 0 mana floating), for 100 separate layers. And in most cases, the numbers being dealt with will be far greater than 100 anyway.
1: "Draw" and play Karakas, exile Simian Spirit Guide and Shinen of Life's Roar to waste the Kira protection on Elesh Norn, then tap Karakas to bounce it. 5 cards in hand.
2: "Draw" and play City of Traitors, tap it, and exile another Spirit Guide to play Daru Sanctifier face down. 3 cards left.
Pitch a spirit guide and tap Karakas to turn Sanctifier face up. Choose to destroy the Nevermore that's naming Mana Clash. 2 cards left.
Pitch a fourth spirit guide to play Mana Clash. Flip 21 straight heads, against 20 tails and a heads.
Given the privilege of just one face up spell, Aura Barbs wins easily against this opponent without needing to manipulate 42 coin flips, but it also needs more mana.
Fork effects are not as useful as additional Rites and other variants like Twinflame. With a Fork, all you do is create a single copy of the spell, and that copy was not "cast" so it never picks up any Precursor triggers. A second copy of the token spell does more than all of the Forks put together, with the exception of getting Eye triggers, a purpose for which they can defer to any blank instant or sorcery: 2 Rite + 39 blanks is better than 1 Rite + 40 Forks. 40 token spells, of course, would be better still, if you're just going for a brute force "do one thing over and over, one card at a time" routine.
Kicking the Rite is also not really a big deal--in fact, given that you have at least 3 copies of Doubling Season (which you will), and at least one token copy of everything important produced by some other means, a mere Sundering Growth with its two-mana cost (that you don't actually have to pay, of course) works out to produce more tokens than paying 5 or 9 mana for a Rite. Yes, even though you are destroying all of your artifacts and enchantments, over and over again. The trick is that Populate is a non-targeted instruction that allows you to produce a token of your most valuable permanent, even on the copies that are nominally "targeting" something else. Here, almost all of the populate steps should go after Doubling Season; you only need to make tokens of anything else once each per round of Precursor Golem spell copies, and only just in time to protect their last surviving specimens from being destroyed by making more tokens that will persist into the next round of copies.
Eye of the Storm and Precursor Golem are also not as much of a combo as would be ideal. You only get a number of Precursor triggers equal to the number of Precursors that exist at the moment the spell becomes cast, and eye of the Storm has you casting multiple spells in succession in the middle of the trigger's resolution, where there is no way to amp up the number of tokens in between casts. You can get a more effective way of scaling by doing away with Eye of the Storm entirely, thereby letting your spells go to the graveyard, then using a sequence of spells that can get each other back from the graveyard (but only in one direction). For example, you could use Sundering Growth to make your tokens, then get that back with Reborn Hope, get Reborn Hope back with Revive, and get Revive back with Mystic Retrieval.
In addition, to get any benefit of scaling at all, you'll want some way of dynamically creating copies of those spells. They don't target permanents, so Precursor Golem doesn't contribute anything at this level, but one neat thing you can do is add an animated Pyromancer Ascension, which you will of course be making token copies of, or else Mirror Gallery + Mirari. In the latter case, you need a source for repeatable generation of mana, since it costs 3 mana per copy of the spell. In the former case, you need ways of "powering up" your Ascensions, because they don't do anything without counters on it, but clearly with Doubling Season you only need 1 trigger per copy to leave it with a lot more than 2 counters. There are a couple different ways of doing this: you could play Mystic Speculation once without buyback, then play another copy with buyback, and for 2 mana you can immediately power up every copy of Ascension that currently exists on the board. There are also a few spells that can return to your hand for free, allowing you to avoid mana altogether: Petals of Insight (although this can potentially interfere with your deck setup, allowing you to go infinite if you have a way to put cards back in the library), Molten Birth, View from Above, and potentially even Titan's Revenge for 0 if you can lose the first clash (or get the initial copy of the spell countered for an illegal target) but then win all the rest. A caveat here is that if you do use spells like Reborn Hope, you can't have any cards that are capable of countering them (such as Odds//Ends), or else you can put it in the graveyard in time for a copy of the spell to return its original self, quickly going infinite.
There's clearly some space for having a few nested operations among instants and sorceries, but the trouble is it's rather limited, and leaves you with a ton of superfluous deck space that isn't appreciably helping. Being able to use that deck space on permanents that have more room for granularity than the broad "Sundering Growth, make lots of copies of everything" will just be more powerful.
2: "Draw" and play Karakas, tap Tolaria targeting Elesh Norn and triggering Kira (oh no, we don't get to make it lose bands with other!), then tap Karakas to bounce Elesh Norn
3: "Draw" and play City of Traitors, tap it for 2, exile a spirit guide, and play Daru Sanctifier face down (which survives, now that there's no Elesh Norn). 5 cards in hand.
Exile a second spirit guide and tap Karakas to turn Daru Sanctifier face up. Choose to destroy the Nevermore that's naming Echoing Truth. 4 cards in hand.
Exile a third spirit guide and tap Tolaria to play Echoing Truth on a copy of Nevermore; all of them will be bounced. 2 cards left in hand.
Play Black Lotus followed by Timetwister, and the rest is easy. The card draws will be replaced by Uba Mask, but you still have access to them for the turn, and nothing's stopping you from playing them.
With the first trigger, you grab Viscera Seer and Body Double, where Body Double copies Protean Hulk. Sacrifice the Body Double and you get to fetch another set of creatures, this time Mogg Fanatic (or Death Cultist) and Reveillark. Sacrifice Mogg Fanatic to its own ability for 1 to the dome, then sacrifice Reveillark to Viscera Seer just to get a trigger. Now you get to reanimate two creature cards with power 2 or less; choose Mogg Fanatic and Body Double, and this time have the Body Double copy Reveillark. From this point, you can keep sacrificing and reanimating both of those creatures forever, or at least until the opponent loses.
Obviously, since this loop can be repeated without limit, it isn't eligible for consideration in the task at hand. But it does go to show the power of recursion: in this case, being able to use Protean Hulk's ability twice means you can get a more potent set of creatures than if you only used it once. Just plonking down a huge swath of creatures and attacking for 100 or so is nowhere near as effective as building a deck around repeating some operation many times. The growth rate of each repetition is also a factor: in the naive 104-damage package I listed, each additional creature only provides a linear contribution of 4 damage. Meanwhile, if Hulk allowed you to fetch 7 mana worth of creatures rather than 6, you would be able to get Hellraiser Goblin (the cheapest fetchable source of haste) alongside Goldnight Commander, which causes each creature to boost the entire army--a quadratic growth rate, such that you could attack for 882, or even more if you decided to add Shield Sphere or the various X-mana creatures that "come into play dead". Even quadratic growth can be made to look puny within the confines of Magic: the most-cited example of a fast-growing mechanism is Precursor Golem + multiple copies of Rite of Replication, although in some cases a two-mana Sundering Growth is capable of exploding even faster than a nine-mana Rite, and there are mechanisms that can get even faster still. You just have to know where to look, and a game with over 15,000 pieces provides an awful lot of possibilities to look through.
That Disciple of the Vault package is far from optimal for a number of reasons: it never causes any "damage" at all, only life loss; the life loss is in increments of 1 at a time so the game will end abruptly while 28 triggers remain unresolved on the stack; and you've only used 4 of the allotted 6 mana. For instance, you could focus on the 0-mana creatures that do stay alive: 4 each of Ornithopter, Phyrexian Walker, Memnite, Dryad Arbor, Crimson Kobolds, Crookshank Kobolds, and Kobolds of Kher Keep (28 creatures--you could also add Shield Sphere but it wouldn't do any good here since it has defender). To go with that, add Hellraiser Goblin (3) and Mirror Entity (3). Tap all four Dryad Arbors for mana since they now have haste, pay for Mirror Entity's ability at X=4, and swing with the 26 untapped 4/4 creatures for 104 damage. And still these two- and three-digits numbers don't get you very far in the grand scheme of things.