Sure you can hunt with it, and sure you can hunt successfully with it. I support your right to hunt with it, plink with it, or do whatever is legal with it.
That being said, MANY (not necessarily all and not necessarily the OP) rely too much on a quick follow-up, and don't take as much care with the first shot. I have seen it many times by many different hunters. This increases the chance of a miss or a non-fatal hit. When you know you have only one, or you will have to manually cycle the action to get a follow-up, you take more care. It's just human nature.
Secondly, no matter how fast the follow-up, the game is most likely to be on a dead run by the time you realize you need a follow-up shot. Taking one in that situation is more likely than ever to result in a miss or a non-fatal hit.
.223 is of marginal power for deer-sized game, as is 7.62X39. I'm not sure what percentage of the AR hunters out there use .308 or above, but the rest absolutely do not have better hunting weapons for deer-sized game than a bolt, lever or single shot in an appropriate caliber.
Other than shotgunning for multiple flying birds, I know of no hunting where a quick follow-up is likely to be of any advantage, except in what should be the very rare circumstance of a bad first shot followed by a game animal still standing there.
Again, I support anybody's legal right to own and hunt with anything legal. To make a blanket statement that AR's are better than manually operated actions for hunting ignores a host of variables in favor of one (perceived) advantage: the quick follow-up to a miss or non-fatal first shot.