I'll be the first to step out on a limb here. If you're looking for one rifle that would cover just about everything at just about any conceivable range then go with a .375 H&H. It will handle anything in North America including all the big bears and will easily handle most everything in Africa and other parts of the world. Although it's a little expensive, ammo is fairly easy to find.

My next choice in a "do-it-all" rifle would be quite a step down but would be a .300 Win Mag.

If you handload the 30-06 will do well though. However I personally wouldn't shoot at anything much over 300 yards with it and 400 would be an absolute maximum. The 375 will be good at that range and them some if you are able to accurately compensate for drop and drift.

As far as why the 308 is so popular with sniper groups is that snipers aren't concerned with killing a 400 pound bear or 1000 pound elk or even a 200 pound deer (though the 308 would work for all of those). They are only concerned with the effect on a human being. A 200 or even 300 pound man is a lot easier to bring down that any wild animal.

I hope this helps. If my only 2 choices were the 308 or 30-06, I would take the '06 any day and twice on sunday.