A gun to the head, for a large animal, is a humane way to end life when done properly. I think it's a matter of personal preference and experience. There are a lot of euphemisms for intentionally ending the life of an animal/pet and I think it’s largely down to the term you used or was most used around you growing up. People that grow up on a farm largely use the terms “put down,” and, much less gentle, “dispatch.”
"Put down" doesn't bother me, and in fact my husband and I have a running joke when any of our dogs acts up where we say we’re going to put them down and then proceed with silly insults, like, ‘your ears are too short,’ etc., as another application for the term “put down” is “to insult.”
“Put to sleep” annoys me, personally – as it’s too flowery for what it is. Euthanize also seems like it's just a kind word for killing something.
You can’t control what phrases or things bother or annoy you – but I think it’s a bit unrealistic to ask people to use a term that only you personally approve of and agree with without regard to the experiences that might make up someone’s particular vernacular.