Notice that's exactly how I worded what I wrote. The constitution doesn't protect those things.
I'm going to put up a giant movie screen right outside your house and play gay porn with the sound system cranked up all the way.
You Cannot do whatever you want with your property.
You said that food is the proper function of animals. You have your own brand of what is proper. I have mine. Animals can be a lot of things ones like dogs evolved as a symbiotic relationship aiding humans. Not as food. You aren't against the branding of purpose on animals just against people who don't agree with your band.
You also seem confused about property rights. Wild animals are not personal property any more than public land is. You don't own a wild turkey, even if it wonders on your land. I can't go and hunt all the wolves in the national parks to extinction any more than I have the right to burn down all the forests to keep warm. Trees in the forest aren't your personal property.
You are also confused about fruits of labor. Not allowing you the fruits of labor in hunting would be allowing you do hunt and kill, but not eat or otherwise enjoy the animal. They are preventing you from doing the work, not enjoying the fruits.