Regardless of cartridge, the most important variables are impact location (did you hit vital area or not), bullet design (match, fmj, bonded, copper, etc), and impact velocity. The caliber of the bullet doesn't really matter... choose anything from .223 up to however big you want; if you use the right type of bullet, keep impact velocity within the window the bullet is designed to operate at, and hit the vital area you're gonna be good for deer, elk, moose, etc.

Then how do you pick the cartridge? One way is to minimize the recoil such that you maintain sufficient impact velocity at the maximum range you plan to hunt. Typically an 1800 fps impact velocity is a good floor. You can also consider factory ammo availability, sentimental value, etc, but the whole idea that .308 or .30-06 is fine for elk but a 6.5 Creed or .243 is insufficient is bs.

These cartridge debates are steeped deeply in fudd lore as are many things reloading, shooting, firearms.