The animal is a pest. Meaning it hosts pestilence, like Lyme disease. The more deer, the more Lyme disease. And that's just one disease. Lot's of wild animals get rabies. There is no cure but eradication.
The animal is destructive. Eats a farmers crops or preys on raised animals. Or, in the case of wild hogs, destroys the land through digging or rooting.
The animal is dangerous. I'm sorry, we eradicated alligator, bears, venomous snakes, mountain lions and wolves for a reason. They can and will kill humans. These type of animals do not belong anywhere we're people are thickly spread, and rural folks who encounter them should not be prosecuted for killing them if they have reason to believe their own life is in danger. Those who say this is acceptable rather than have the animal go extinct are not talking about themselves or their own kids getting eaten, however. Just "other" people--not them. I'd prefer such people themselves go extinct. Hopefully by a wolf, cougar, bear or alligator. Karmic!
Leather or fur. I've tried gator--the tail. I wouldn't go out of my way to eat it though. But alligator leather is top-notch. I like fur, but wouldn't eat a Mustelidae like a mink or skunk because that is some nasty stuff!
I ate squirrel - that was enough for me. Not a fan of venison but like you said, they are pests. Many hunters who enjoy the hunt but don’t eat the meat, donate it to local food pantries. So all is good.