Is one allowed to kill an insect on Yom Tov that is "bothering" them (causing some sort of tzar)? (I'm inclined to think, based on the Halachic concept of "m'toch", that since we are allowed to shecht (slaughter) an animal in order to eat, so too the action of killing something would be mutar for other reasons as well. Is this correct?)
Lately, I've heard a lesson from R. Avrohom Kuperman about this topic (what a coincidence!), and he finally told that it is forbidden.
The line of thinking that it should be permitted (as you presented it in the question) is correct, but! We only allow to make melachot (actions) that you enjoy from the melacha itself. But in this case of killing insects you are just remove the disturber and not enjoy from the body of the insect.
This was a long and interesting lesson, and final proof he brought is that Mishna Berura (תקלג, ס"קכ) allows to kill insects on Chol hamoed. From that he concluded that on Yom Tov it is forbiden.
The M"B in simon 627,30 writes that one is not allowed to carry a dead body out on yom tov, even if having a dead body in your house bothers you, even if he is a kohen and is therefore not allowed to go into his own house, even though carrying on yom tov is one of the malachos that we say mitoch, the reason for this is because we only say mitoch on a positive action, not to get rid of something (siluk) based on this one should not be allowed to kill bugs on yom tov. (and take out the garbage!)
In addition according to allot of poskim (ravyah, Ohr Samech, etc) when one kills a big, he also trapping the bug, trapping is not allowed on yom tov