Well for once i'm going to somewhat agree with Mikey, at least as far as this post goes.
For most of what the OP wants to accomplish, the 308 would be a good choice, especially for the hunting part due to bullet selection.
Along with the 06, they have been being used for 1000 yd competition at places like Camp Perry for a very long time.
We use them for practice shooting at rocks at our camp even further than 1000 yds.
My grandson has a Savage model 10 FCP in 308 and i think its the most accurate factory rifle ive ever seen.
We dont shoot for groups, at all, ever, in fact i could actually care less about small groups.
Our interest lies in the rifle being able to repeat hits on the rocks we shoot at at the distances we hunt.
And most importantly, us not losing the target due to recoil, and scoring a second round hit if the first one fails, regardless as to how much.
Using the hammer is the issue, at least to a point, not the actual hammer itself.