Couple of things that come to mind.
First, all these hunters who plan to just go whack a deer, do so the first week. How many can go beyond processing and cooking the large amount of meat they suddenly have?
So how much will spoil before they can eat it all?
Out they go again for another, pretty soon theres few to be found, and decades to rebuild the herd.
Soon starvation begins,
Two: How many people can actually find a deer within walking distance of their home? Depending on what actually happens cars may not work, be it EMP or just not being able to afford gas.
City folks especially can dream about heading to the hills and taking all the game they want, but the reality isn’t so bright.
Third: Even small game have cycles, boom and bust. Those rabbits that one may depend on to fill the pot, may not be around either.
Then there’s rabbit starvation. Unless you have a means of adding fat to your diet, you can and will have issues even though you are eating.
As far as the rustling of various farm animals, theres a lot of ranchers that might take offense with that, and would likely treat anyone trying to do so as any other predator.