The Springfield M1A is designed for shooting the higher pressure .308 Winchester cartridge. Older Mausers and other actions that were converted 40 years ago were proofed for the 7.62x51 loading. IIRC the .308 Win has about 5,000 PSI more maximum allowed chamber pressure than the 7.62x51 and the 8x57 cartridge that most converted rifles used before they were rechambered.
While pressure is the popular theory on old rifles and .308, I believe it is incorrect. Both rounds will chuck the same weight projectile at the same speed, I don't think we're seeing much of a pressure difference. The confusion comes from the two different systems each uses to measure chamber pressure. The real problem, I believe, is chamber length.
Lets look at the numbers for chamber gauges for the two calibers:
.308 GO 1.630"
.308 NO-GO 1.634"
.308 FIELD 1.638"
7.62 GO 1.6355"
7.62 NO-GO 1.6405"
7.62 FIELD 1.6455"
Among the several differences there, notice that the nato field gauge is significantly longer than the .308 gauge? I would personally put a .308 field gauge into any milsurp I wanted to shoot .308 through. If it closed on a .308 field gauge, I wouldn't shoot .308 through it. I would then check to be sure it was even safe to fire nato in it.
A gun like a 700 has its chamber cut for the .308 lengths so it shouldn't be an issue with either. Military guns chambered for nato, I'd test.