the majority of the population doesn't know how to hunt, and will not be very successful.
Those that do know how to hunt will do ok, but game will seek out places that are very hard to get to. In tough times most people are not going to be chasing game at high altitudes or through very tough terrain. They also won't be traveling through desert and other areas. Game that lives on the tundra or other extreme climates won't be overly hunted either. I suspect that if some big SHTF scenario does unfold, people such as the Inuit will be unaware / unaffected.
There will be game for those who know how, and have made the right preparations. I'd go after game BEFORE using up my preps. This gets you the easy stuff before it all becomes educated and heads for the tougher areas. Then later on, I would still be in better shape food wise and in better physical shape if I wanted to pursue into rougher country.
Starting out relying on game is a bad idea. However using it early on, if possible, to stretch your preps is a good idea. worse case you have to rely on what you have on hand. Best case you get to supplement what you have and make it last longer.
This is one of those things that I have to file in the 'dammifino' file. Superficial logic would seem to say 'yes', but Dorobuta's correct that most people don't have the skills to hunt. So will those people take game, will they wound and waste game, or will they just scare off game out of their area..?
On topics in the 'dammifino' category, my default is to try & come up with a historical precedent. IE, "under what circumstances, if any, has __ happened in human history?". So in this case, has game completely disappeared where hunting became an absolute non-issue? I can't think of one. Gotten harder or scarcer, sure. But "gone" completely? It's possible, but I'm not aware of any. That said, it could absolutely be scarce enough that I wouldn't want to count on it. (And even as crazy rural as we are, I still don't count on it.) Even now, in normal times, I don't want to try to support us food-wise by hunting. So it wouldn't make sense at all for me to want to do it in bad times.
One more reason to not be the backpacking wanderer of the apocalypse. If you can't count on being able to find food, it means you have to rely on stored food; and storing food requires a home base of some form.