All of the cartridges that have been discussed here, in fact most military calibers over 6mm are adequate for most North American game, although the 6.5s might be a little light for stuff in the elk and moose range.
For a reason other than the cartridge I think that there are only a few that are suitable hunting candidates. The best choices in my mind are the Lee-Enfield (No1 or No4), The P-14 or M-1917, any 98 Mauser variant, 1903 or 1903A. The reason is the way the safeties function. All of these rifles have safeties that can be easily and quietly disengaged. That means the safeties will get used.
The safety on Mosin-Nagants, Schmidt Rubins, Carcanos and Arisakas are difficult to use in a hunting situation, that means you are more likely to be carrying it with a round up the spout and the safety off, meaning an accident is more likely.