Well bear hunting is not as dangerous and man-hunting, and our military uses Auto-loaders for that. So does every other military on earth.

Most of the dogma of how reliable a double rifle is came from writings of 85-150 years ago. And at that time doubles were the most reliable hunting arm available. But I would point out a fact here;
Doubles were reliable in the hunting field BECAUSE if and when they broke you still could hunt using the other barrel. It was like having 2 rifles. So in that knowledge we must define the word “reliable”. 100 years ago “reliable” seemed to mean a weapon you could still hunt with even if it broke. Today “reliable” seems to mean a weapon that doesn’t ever break.

I hunt with a flintlock much of the time. From a standpoint of use and abuse, it's probably the least reliable gun I could use. (Matchlocks and wheel locks would be even less reliable, but I have not hunting with them) So I am not one that worships at the feet of 100% reliability when it comes to hunting, but as a gunsmith and as a former military man I do understand the issue very well.

When we look to the M-98 Mauser and its copies (with or without some modifications) we find a rifle that truly is more reliable then the double rifle. That bit of news seems to drive a lot of double rifle fans crazy but it is none the less true. If you think not we should look at the “test to destruction” that has been done of various rifles and even on handguns.
The criteria is pretty simple.
Fire the weapon in either lab conditions or field conditions until something causes it to fail in a way that requires parts to be worked on or replaced. This usually doesn’t include the barrel, the failure of which is judged on its ability to shoot accurately, not just to shoot bullets. Bore erosion is a function of the cartridge fired, not the type of rifle it’s screwed into.

Many controlled feed bolt actions have gone over 1,000,000+ rounds. In many cases good bolt actions have not failed before the test ran out of ammo. I have never known a class of double rifles or shotguns to be able to boast of that kind of reliability.

So coming back to the original subject, we need to look at the current autos that are being discussed. The AR platform.
It’s pretty common to see them go through many many thousands of rounds before they jam. When they do jam they usually can be cleared in about the same amount of time as you can reload a double. I work on them a lot in my shop and I will say that they are NOT as reliable as a good controlled feed bolt action, but they are not bad enough to make me reject them as a hunting weapon. They may someday attain to the reliability of the AK platform which is probably the most reliable auto rifle ever made. The regularly DO go thought several barrels with no jams at all. I still doubt that an AK is as reliable as a properly built Mauser system or M-70 Winchester, but I would not say dogmatically that I know it. To be able to say that with certainty I would have to have 5 rifles of each type and enough ammo to shoot them all to a failure (and enough years left in my lifetime to do that test, which I do not think could be done quickly).

To complete that test I believe I would have to fire many millions of rounds to get an average. Then I might be able to say the Average M-70 or M98 is more reliable than the average AK or visa-versa.

What I can say right now and right here is that the average M98 or M-70 is going to be just fine for hunting for the average man today.
I can also say that for any well build AR or AK.
Heck I would not feel a bit under armed hunting big bears with my old WW2 M-1.

In good examples, autos are just fine for hunting.
In bad examples (of which we do have many) no rifle is just fine.