You sound like a friend of mine when I was in College.
He too wanted a 308 for big game. He picked 308 for a couple of reasons but one was it was the prefered 1,000 yard round and the military snipers were using it. 87-92 I forget what year he got it.
We he purchased a hunting rifle and then wanted to scope it with a target scope. (they did not have the tactical rifles back then and the 700 V was about all you could get in that class and pretty heavy for a hunting rifle)
So we go out to my family's farm in NC where a 1,000 yard shot was possible and after sighting in the rifle at 100 yards then shooting at 200, and 500. He was not happy with his groups at 500 yards. He expected the same ragged hole he had at 100 yards. With factory ammo the difference in powder, primers, projectile weight, and the internal case all vary a little that will change you velocity and there fore change you down range impact. While the scope was a little too big for the hunting rifle it was a good hunting rifle and he took a few deer with it. But he was never happy with that rifle or the three or four that followed it. Again these were hunting rifles and not target or varmint rifles set up for long range shooting.
I compare a lot of things to Pick up trucks.
So bare with me on this one.
A pick up is great and can do a lot. But if you are looking to haul big rocks it may be best to get a Dump truck.
The dump truck is specifly designed for one task- hauling haevy stuff. the pick up is a jack of all trades and can do lots of things.
A general purpose hunting rifle is a pick up. It can do most of what you ask but is not exactly right for the specific job at hand.
And the reverse. the Target / tactical/ varmint rifle is great for these tasks but is either too heavy, or to bulky to have as a hunting rifle.
I like 308 and have two. A ruger 77 with iron sights and a Rem 700 V that wears a 3-9X50 scope. Both work for me in what and how I hunt. But the 700 V is too heavy and the 50MM optic is too big for long range hunting or horse back. The 77 light and is great for in the woods and the fields of NC or CA but the open sights limit me for longer shots for pigs and deer in open areas. i veiw the rifles in my gun locker as gold clubs and I would not use a driver to put.
Get a good hunting rifle put good glass on it 3-9X38 or 40. i would stay away from adjustable objectives. and sight it in for 200 yards and use it as a hunting rifle. As you want to get into the long range shooting use the hunting rifle to see if you like it. Others will come over and let you shoot thier rigs. Trust me. then deside on what you want as a 1,000 yard rifle.