The real reason for the short mags was to reduce manufacturing time, expenses and inventory for the manufacturers. They can make more short actions, save a little steel and put out a greater variety of calibers with same/similar actions.
Doing that allowed marketing geniuses to come up with statements that suggested magical performance from a shorter cartridge.
Oh, and charge more.
But, for what you want, I would go with 30-06, especially for the black bear and elk. Close second choice, .308 and you would get a shorter cartridge.
Really, it is a matter of choice but I like the greater versatility of the 30 cals.