To most North American game, the animals you'll likely be shooting couldn't tell the difference between the '06, .270 or the .308!
So I'd take the .308 in any of the guns you listed, and never look back! That said, if I were likely to run into something that would like to have me for dinner, I'd want to have something bigger than any of these three cartridges!
If you are looking for one gun to do it all from deer on up, I would not look at anything less than a .300 magnum! It's overkill on the smaller animals, but you'd be glad to have the extra poweron the dangerous game side of the coin!
When using the guidelines of "ONE gun" AND"ALL North American game animals,"I just would not bet my life on anything less than a .30 caliber magnum or a big bore cartridge!