Alright, first of all, forget the .22 rimfires. Mountain men trapped their small game, they didn't shoot it.
There's two rifles I'd pick from, and I'd be happy with either.
First is a Remington 700 BDL in .30'06. 22" barrel, iron sites (in case your scope goes lenes up in the middle of nowhere, you still have sights), and a very popular and capable caliber. This is probably the more practical of my two choices.
The other would be the much mentioned .45-70, although, after feeling them up in the gun shop, I like the 22" 1895 better than the 1895G because of the pistol grip. I also don't think 3 1/2" makes that much difference in packability. Now there's always two complaints with the .45-70: ft-lbs of energy and trajectory making it a short range cartridge.
The .45-70 doesn't wound by hitting at mach 5. Lower velocity means lower ft-lbs, and that simply cannot be used to compare rounds of different caliber. Sure, it doesn't have much in the way of kinetic energy at 200 yards, or 300, but it will outpenetrate all of the mentioned rounds at that range. The .45-70 plows a big hole in game, at low velocity. This is a good thing because it doesn't destroy the amount of meat a faster projectile will.
The second thing, the trajectory making it short range and supposedly "needing" a long range rifle. The .45-70 is a very accurate cartridge. It's a matter of knowing the trajectory and your holdovers/scope adjustments for a given range. OK, the flaw is that it is not a lazy man's rifle. But if you want to be a mountain man, you can't be lazy in anything. Also, a long range rifle is NOT "needed". Granted long shots make up for hunting skill, but let me remind you that ALL of this game that is supposed to "demand" a long range shot has been historically, and is currently, successfully hunted with bows. A 40 yard shot with traditional archery gear is a long shot. If you can get that close with a bow, you can get within 100 or 150 yards for a shot with a .45-70 (or 12 ga slugs for that matter).