700 BDL in 30-06 will fit the bill perfectly. The 308 is great but it really can't shoot the 220 grain slugs well which is what you should use if you go after Griz, Buff or Moose. The 30-06 is found all over the world, so is the 308 and the 7x57 Mauser. The deal with the 06 is that it really can cover everything from soup to nuts for the average guy and finding ammo is never going to be an issue.
I assume you want a hunting rifle. That negates any advantage the 308 might have in accuracy. You need 1-2" groups at 100 yards. You don't need clover leafs. Theory is one thing but practical application is what counts in a hunting rifle. Any good 30-06 will shoot better than you can under hunting conditions.
The Rem 700 is such a ubiquitous rifle that parts and service are available anywhere on the planet. It's a proven and trusted platform that works under all conditions and it fits the average guy without alteration.
That's my $.02. I'm 54 years old and have hunted big game in AK, MT, ID and CA. I'm a Hunter Ed instructor for the state of CA and a long time rifle nut. I shoot the 06, 308, 8mm Mauser, 7x57, 45-70 and 45-90 along with the 32-20, 357 mag (in rifle and pistol), 375 Win BB, 38-55, 358 Win, 22-250, 223 and of course 22LR. So, I do know my guns and do have some varied experiences with big game hunting. You have my take. Good luck and have fun!