Military ammo is one thing; hunting rifles are another.
For war and for match shooting, the .308 has the edge, if only because of availaiblitly and ease of feeding in auto weapons and commonality with machine guns using the same ammo.
The hunter wanting a ctg. with flatter trajetory may well prefer the excellent .270.
When the .308 came out, W.D.M. Bell was still living and he commented that the short action would have pleased him greatly when he was ivory hunting with his famed .275 Rigby (7X57mm with Rigby brand bullets). Not only is the .308 bolt throw shorter than for 7mm, .30/06, etc., the shorter action allows making a shorter rifle that's lighter.
Nonetheless, if I was looking for a rifle for pronghorn and other plains game, the .270 Winchester would be my first choice. If the plains game gets large, elk size, like the oryx or the sable antelope, it still works, as Jack O'connor proved. But I think I'd lean toward a .375 H&H Magnum for them. But O'Connor did find the 130 grain Nosler bullet in .270 worked well on lung shots. He also found that it killed moose as well as anything else he tried.
I think you should find and read,"The Rifle Book" and, "The Hunting Rifle" by O'Connor before buying a sporting rifle.