The strategy is to quickly ramp in mana through defenders, Primeval Titan, and Sylvan Primordial. Use this mana advantage to get out big creatures and support creatures early and control the field. Use dispel and negate to keep alive your support cards such as Prophet of Kruphix and Arcanis the Omnipotent, as they keep you fueled and put your opponent in a tough position.
I don't mind the idea of primeval titan or sylvan primordial in a deck, but I won't lie: you're going to have a tough sell running them in legacy, where any card above cmc 3 had better win you the game. Most of your stuff, unfortunately, does not do that. You have precious little interaction with your opponents and you don't ramp fast enough to ignore them. That said, the strategy can work with the right cards. Most of them aren't so budget, though. Check out the blue/green 12 post thread in developing competitive for ideas there. If you want to pursue the strategy, then things to keep in mind:
a 5 mana 2/3 will not do well in a competitive meta, edh aside. If you're going to run a 5 mana creature in the prophet of kruphix slot, then the only one really worth considering is teferi, mage of zhalfir. He has flash himself, and he completely stops opposing spell based interaction. I still would be hesitant to run him even, and then not more than two.
Similarly, arcanis the omnipotent is a fun card, but at 6 mana, it doesn't end the game fast enough and is most assuredly not worth a 3 x slot in. I would run zero, or at most, 1, and that is only *if* you run a tutor engine to search him up on the off chance he'll matter in a game. Unfortunately, the best search engine for this slot is green sun's zenith and arcanis is not green.
Sylvan primordial is quite an entertaining answer to non-creature problems. However, he's highly costed and does little enough in the 1 one 1 environment that is legacy. He's flat broken in multiplayer, but if I were going to try something like this, then I would rather see diluvian primordial. Legacy and vintage are the formats of game breaking spells, not as much non creature permanents. Even so much as sniping a lightning bolt or swords to plowshares on an opponents creature+big 5/5 flier can be a much bigger game breaker. Getting something such as a brainstorm, ponder, or similar is probably going do you well also. Same story as arcanis, though. Only 1 at most!!!
Same thing can be said about engulfing slagwurm. He's fun, but he's not good enough. There is absolutely no guarantee your opponent will have a creature to block with or that he won't just let it through and crack back with his own tarmogoyf. You are correct in thinking that lifegain can help (burn is all over the place). I would suggest kitchen finks or spike feeder. Both block, gain some life, and stall until your big spells matter.
Primeval titan...he's the one creature in your list I have little beef with. He's big, tramply, immediately brings in two useful lands (presuming you have at least something you want to get besides basics...personal favorite is wasteland, but there exist many less expensive options to go get. Again, see that thread I pointed you to earlier for obnoxious ideas). He could also fetch something as innocuous as two glimmerposts to gain life and stabilize.
Now, possible solutions to actually MAKE this somewhat competitive:
You have precious little interaction. You run no land based disruption which would justify things like primeval titan or sylvan primordial. Your support spells will probably protect your beaters well enough, but unfortunately, they probably won't stop your opponents from resolving game ending threats until you can get to your beaters to be worth protecting. The last thing this deck wants to see is a humility or jace, the mind sculptor, and until turn 4 or 5, at earliest (presuming you've lucked out on tons of mana generation), you can't out card advantage them (arcanis being that outlet). On top of that, you have no real way of turning late game dead draws such as axebane guardian into something useful, or at least exchanging them for cards that might be useful.
I will say that you probably have a solid early game against very small aggro decks thanks to 10 defenders. That won't matter diddly to a burn deck, though.
For stack disruption, you want the original, the best, 4x counterspell, no questions asked. Just "stop that" written all over it. Negate and dispel are good counters, but ultimately, you don't have enough of them and they aren't general enough to justify their numbers in this particular deck. I would check around other blue based threads to see what non force of will counters are worth looking at.
For other styles of disruption, some land based removal such as bojuka bog or the glimmerposts might be worth looking at for a primeval titan deck (which is where this really wants to go..insurmountable late game advantage that shuts it all out).
Otherwise, there are other styles of disruption. Maybe back to basics or root maze are worth considering? The first one slows down many, if not most, opponents by a ton and the latter slows you down. It might be worth it for your deck, though, as other styles of deck can outright kill you.
Maybe aggro decks give you fits early on and you need to take some extra turns to stabilize. Moment's peace seems a great add in here. It doesn't give them a creature and gives you time to make a really big threat that they can't deal with. Even better, it does it twice for one card.
Card selection? Brainstorm turns late game dead draws into something more useful, or maybe ponder or preordain are good ideas. Either way, they are early card selection which most blue decks want, and you lack little of. The ability to find your limited answers is rather necessary, ramp deck or not!
Other than things I mentioned, this deck can be fun to play and considerably more competitive than it is. However, I am not going to sugarcoat it and say you'll do well at some big event: Likely, you won't. But you can tune it do take some games/matches at your local FNM level meta.
Also, have you considered the possibility of utility chains from big to little creatures? It'll be something unique to your style of deck, but it could be fun. As an example, woodland bellower is a card that's pretty cheap and finds lots of utility creatures from eternal witness to kitchen finks to reclamation sage to your defender ramp to even a singleton off-beat suggestion of fierce empath (to chain up another primeval titan or woodland bellower).
Creatures (28)
Overgrown battlement x4
Axebane Guardian x4
Sylvan Caryatid x2
Primeval Titan x3
Sylvan Primordial x4
Engulfing Slagwurm x3
Spearbreaker Behemoth x1
Prophet of Kruphix x4
Arcanis the Omnipotent x3
Spells (8)
Negate x2
Dispel x3
Reality Shift x3
Lands (24)
Island x11
Forest x11
Simic Guildgate x2
Sideboard
Whirlwind
Back to Nature
Lure
a 5 mana 2/3 will not do well in a competitive meta, edh aside. If you're going to run a 5 mana creature in the prophet of kruphix slot, then the only one really worth considering is teferi, mage of zhalfir. He has flash himself, and he completely stops opposing spell based interaction. I still would be hesitant to run him even, and then not more than two.
Similarly, arcanis the omnipotent is a fun card, but at 6 mana, it doesn't end the game fast enough and is most assuredly not worth a 3 x slot in. I would run zero, or at most, 1, and that is only *if* you run a tutor engine to search him up on the off chance he'll matter in a game. Unfortunately, the best search engine for this slot is green sun's zenith and arcanis is not green.
Sylvan primordial is quite an entertaining answer to non-creature problems. However, he's highly costed and does little enough in the 1 one 1 environment that is legacy. He's flat broken in multiplayer, but if I were going to try something like this, then I would rather see diluvian primordial. Legacy and vintage are the formats of game breaking spells, not as much non creature permanents. Even so much as sniping a lightning bolt or swords to plowshares on an opponents creature+big 5/5 flier can be a much bigger game breaker. Getting something such as a brainstorm, ponder, or similar is probably going do you well also. Same story as arcanis, though. Only 1 at most!!!
Spearbreaker behemoth needs to go. The only spells that see common play that are going to remove your beaters in a 1 on 1 fashion are swords to plowshares and path to exile and neither cares about indestructible.
Same thing can be said about engulfing slagwurm. He's fun, but he's not good enough. There is absolutely no guarantee your opponent will have a creature to block with or that he won't just let it through and crack back with his own tarmogoyf. You are correct in thinking that lifegain can help (burn is all over the place). I would suggest kitchen finks or spike feeder. Both block, gain some life, and stall until your big spells matter.
Primeval titan...he's the one creature in your list I have little beef with. He's big, tramply, immediately brings in two useful lands (presuming you have at least something you want to get besides basics...personal favorite is wasteland, but there exist many less expensive options to go get. Again, see that thread I pointed you to earlier for obnoxious ideas). He could also fetch something as innocuous as two glimmerposts to gain life and stabilize.
Now, possible solutions to actually MAKE this somewhat competitive:
You have precious little interaction. You run no land based disruption which would justify things like primeval titan or sylvan primordial. Your support spells will probably protect your beaters well enough, but unfortunately, they probably won't stop your opponents from resolving game ending threats until you can get to your beaters to be worth protecting. The last thing this deck wants to see is a humility or jace, the mind sculptor, and until turn 4 or 5, at earliest (presuming you've lucked out on tons of mana generation), you can't out card advantage them (arcanis being that outlet). On top of that, you have no real way of turning late game dead draws such as axebane guardian into something useful, or at least exchanging them for cards that might be useful.
I will say that you probably have a solid early game against very small aggro decks thanks to 10 defenders. That won't matter diddly to a burn deck, though.
For stack disruption, you want the original, the best, 4x counterspell, no questions asked. Just "stop that" written all over it. Negate and dispel are good counters, but ultimately, you don't have enough of them and they aren't general enough to justify their numbers in this particular deck. I would check around other blue based threads to see what non force of will counters are worth looking at.
For other styles of disruption, some land based removal such as bojuka bog or the glimmerposts might be worth looking at for a primeval titan deck (which is where this really wants to go..insurmountable late game advantage that shuts it all out).
Otherwise, there are other styles of disruption. Maybe back to basics or root maze are worth considering? The first one slows down many, if not most, opponents by a ton and the latter slows you down. It might be worth it for your deck, though, as other styles of deck can outright kill you.
Maybe aggro decks give you fits early on and you need to take some extra turns to stabilize. Moment's peace seems a great add in here. It doesn't give them a creature and gives you time to make a really big threat that they can't deal with. Even better, it does it twice for one card.
Card selection? Brainstorm turns late game dead draws into something more useful, or maybe ponder or preordain are good ideas. Either way, they are early card selection which most blue decks want, and you lack little of. The ability to find your limited answers is rather necessary, ramp deck or not!
Other than things I mentioned, this deck can be fun to play and considerably more competitive than it is. However, I am not going to sugarcoat it and say you'll do well at some big event: Likely, you won't. But you can tune it do take some games/matches at your local FNM level meta.
Also, have you considered the possibility of utility chains from big to little creatures? It'll be something unique to your style of deck, but it could be fun. As an example, woodland bellower is a card that's pretty cheap and finds lots of utility creatures from eternal witness to kitchen finks to reclamation sage to your defender ramp to even a singleton off-beat suggestion of fierce empath (to chain up another primeval titan or woodland bellower).
Credit to DolZero for this awesome sig!