The main reasons I pick a .357 is ammo — .38 and .357, plus a good wheel gun can sit around forever (assuming well taken care of) and still go bang. It would also work on really crappy reloaded ammo if the world really fell apart.
I keep a .22LR and .357 under the spare tire in every vehicle I own, with a holster, mags, tennis shoes/socks, $1,000, etc.
You could hunt OK with any of the ones I listed, and all of them are fatal.
If I had to pick ONE, I’d pick the Ruger 10/22, mainly because you could pack a bag with 1000 rounds and it would last you years in the woods.
The AR-15 trumps the shotgun (barely) for this reason, as well as magazine capacity and distance accuracy.
And with a Judge you can use .410 snake-charmer rounds.