Some people have trouble killing big game with large calibers and some people routinely kill any animal with small calibers. Woman and children seem to be particularly adept at killing just about anything with a 22 or 24 caliber. As you stated, eskimos do it everyday all day. They say "Too small for a white man but just right for an eskimo". 22 or 24 is just fine for polar bear, moose, grizzly, or walrus.
The difference is in the person, not the caliber. The caliber has the ability, the person not so much. A lot of big game animals are killed with bad shots. So many animals are killed with hits on the rear of the lungs or liver because the shooter, intentionally or not, aimed mid mass. A lot of animals are spined but the shooter was not intending to spine the animal. Arteries are hit under the spine or even in the rear leg although they were not aiming for those areas.
Some people have the ability to kill with the skill and coolness of an assassin, others get flustered by adrenaline, not sure of their precise aiming point, in the overload of the moment disregard essential markmanship skills, jerk triggers, flinch, and make comments like you don't feel the recoil when shooting at an animal justifying using a caliber they have proven to themselves they can consciously control at the bench, but not when combined with the numerous other stimuli in the field.
Shot placement is really what matters, plenty of 22 caliber bullets to do the job. As a bowhunter with 1 shot I have to very cautiously choose my angle, as a muzzleloader with only 1 shot I carefully wait for the right shot, as a turkey hunter I wait till the head is up and clear, as a rifle hunter no reason I can't be somewhat disciplined to make the correct 1 shot kill also.