Being able to kill an animal with minimal remorse is something you can learn.
I used to be a vegetarian for most of my adult life (not for reasons of animal welfare, more out of protest against the farming industry). Since my wife and I started a modest little farm with chickens, ducks and a few cows, I started to eat meat again on a regular basis. Still not the amounts most people eat, but I enjoy eating what I've raised more than anything else.
For me it's a click in the head: I know this or that calf will be butchered some day, and if I were allowed, I would rather kill it myself than let a stranger do it. Not because I have grown a liking to killing animals, far from it, but just for the sake of the animal. Being in a familiar environment or with a familiar person when it happens, rather than being put in a stressful environment with strange people surely helps to reduce suffering to a minimum. It's a matter of respect. With wild animals, they are never/always in a stressful environment, so shooting it doesn't add to that unless you don't shoot straight...
As for pain: even lobsters feel pain, as suggested by research
http://blogs.nature.com/news/2013/08/experiments-reveal-that-crabs-and-lobsters-feel-pain.htmlSo it's a hunter's job (and a farmer's and butcher's job) to ensure that suffering is reduced to a minimum. Just out of respect for the animal you will eat or sell.
And there's a final economic argument as well: the meat of stressed animals is less tender and tasty than that of unstressed animals. So it's also in the farmer's own interest to stress his animals the least possible. And probably also in the hunter's interest to have a clean quick kill (apart from the time not spent trailing a wounded animal).