Most of my Ruger rifles- including a pair of Americans- did not require much shooting at all to reach their full potential. A couple grouped just as well out-of-the-box as they did after hundreds of rounds. The others were fine after no more than a couple dozen rounds. And one of mine in particular needs to be "dirty" to shoot well. My 300 Win Mag 77MkII All Weather needs 20 or 30 down barrel to foul it before it groups well. That one really messed with me...I thought is was garbage when I first got it, because it only did around 2MOA with its favored load. Of course I cleaned it after a range session, which typically was 20 rounds or less. One day, being bound and determined to find a good load or send it down the road, I took a bunch of new combos to the range and shot 60 rounds. After 30, groups started to shrink, until I was getting just under or just over MOA with virtually anything I wanted to send through it. Weird-azz gun...
If one doesn't handload for the .308, I recommend getting a box of Federal match ammo and trying it after verifying all screws are properly torqued, both action and optics. If it doesn't group well with that, there is either an optic issue or the gun is a pooch. (If one handloads, the Sierra or Nosler 168 grain BTHP match bullet over 46 grains of Varget has never failed to group well in a wide variety of rifles for me.)
Another thing to look at is the parallax of the scope. If it is fixed, at what range is it parallax-free? If it has an AO, are the markings on the dial accurate? I had a friend (unknowingly) mount a fixed objective scope designed for rimfires on his 30-06 and initial sight-in at 50 yards was just a small ragged hole, but he had a hard time hitting a 12" plate at 300. The scope's parallax was fixed at 50 yards, and the slightest misalignment of his head behind the ocular resulted in severe deviation of the bullet from the crosshairs at 300.
The stocks on Ruger Americans can be pretty flimsy. If you (like I tend to do) muscle the rifle on the rest, the forend can flex and apply pressure to the barrel, causing flyers. A light hold may be neccessary.
As for bullet weights? I shoot 180 grain Ballistic Tips or SSTs for hunting, and they group like match ammo from my 308s (a Model 70 Heavy Varmint and an LR308). They also perform exceptionally on deer.