So....the emphasis is on caliber, not rifle. You kinda asked the same question in two different ways.
Don't know that accuracy and "all aroundness" go hand in hand all of the time.
If you are shooting only factory ammo, I'd go with the 30-06: proven accuracy, lots of different types of firearms, a gazillion different bullets - from 100 grains up to 250 grains. I'd pick it over the .308 only because it will handle those heavy bullets better.
As a handloader, I'd pick the 375 H&H. I can load it down and shoot round balls at squirrels, duplicate the .375 Win at any level, or load it all the way up and take it out to Africa or North for the big bears.
For true "all aroundness", though, wouldn't the firearm have to be able to do some wingshooting? Maybe a rifle isn't the answer - maybe the best choice is a shotgun; small game, big game, self defense, clay games.
Pete
PS - Maybe even the 45-70.