That is assuming your opponent is not in the state of killing you*
In your example, if your opponent is in lethal range of your pestilence demon activations, and you activate it enough times to kill them, even if they kill the demon in response to any of those activations, the abilities are already on the stack and they will die.
But if you yourself somehow died, everything you've contributed to the stack are gone. For example if you have less life than that opponent, you will die on the resolution of one of the many pestilence demon activations before your opponent.
There are other effects that may come up here like split second cards that are needed to resolve before more activations, or time stop effects that may disrupt the stack. But those are not too commonly seen.
In your example, if your opponent is in lethal range of your pestilence demon activations, and you activate it enough times to kill them, even if they kill the demon in response to any of those activations, the abilities are already on the stack and they will die.
But if you yourself somehow died, everything you've contributed to the stack are gone. For example if you have less life than that opponent, you will die on the resolution of one of the many pestilence demon activations before your opponent.
Killing opponents is the best counterspell.
There are other effects that may come up here like split second cards that are needed to resolve before more activations, or time stop effects that may disrupt the stack. But those are not too commonly seen.
Links to my most current deck lists;
Primary EDH; Rakka Mar Token Perfection, Crosis Mnemonic Betrayal, Cromat Villainous, Judith Gravestorm, Rakdos Empty Storm, Exava Artifacts, Bant Trash, & Fumiko Voltron!
EDH kept at home; Ruzzian Isset & Rakdos LoR!
EDH (nostalgic/pimp/retired) in storage;
Latulla Burns, Akroma Smash, Jeska Voltron, Rakdos Storm, Bladewing Darghans, Lyzolda Worldgorger, Xantcha Steals your Heart, Jori Storm, Wydwen Permission, Gwendlyn Paradox, Jeleva Warps, & Sigarda Brick!
Legacy Showanimator and High Tide!