I really agree with most of your post, the whole thing is exactly what needed to be said. However, as an avid hunter myself I would like to build on the concept:
Population control, in my opinion, is a reason to hunt that developed itself into the sport as development became more prominent. Originally people hunted for food, thats where the idea comes from. With the rise of domestication that need went away. People kept hunting because they sought amusement from it, the thrill of the chase, etc. You have to be careful not to overlook the sporting aspects of hunting, but you also cant understate the importance of hunters to game animals. Its gonna sound fucked up to a lof of people in here but hunters are a critical part of wildlife management beyond population control. We are largely responsable for things like habitat management, through organizations like Delta Waterfowl, The National Wild Turkey Federation, Ducks Unlimited, and Pheasants Forever. All these organizations are key to the existance of game animals in a developing world.