As said, the .308 is a good cartridge. However, cartridges do not fire themselves, and the rifle you use for hunting may not be the rifle for benchrest target shooting.
I have one of these in .308
It is a Remington 600. While it IS a very handy light little rifle, it is NOT a 500 yard target rifle- and it sure is not an 800 yard rifle. However, I do not want to CARRY a benchrest rifle hunting.
A key part of long range shooting is being able to read wind and mirage. A .308 with a 147 gr bullet zeroed at 100 yards will be 57 inches below line of sight at 500 yards. A 10 mph cross wind will shift the bullet about 22 inches to the side.
I am NOT a champion long range shooter, and not a military sniper. Have known my share of them, watched them shoot, and was in awe.
Of course, for really long range work, I preferred 155 mm artillery, and high performance aircraft with CBUs and napalm. Snipers were retail, we were wholesale.