Training issue=
If you want and or need what a 7.62 gas gun offers then you work with the weight and deal with it. There are only so many of them that work well enough to be more than a play thing, and none of them are lightweight.
There is a reason that those who use guns as a job that have both 5.56 and 7.62, overwhelming choose compact 5.56's. For most that juice isn't worth the squeeze.
As far as the original question- no, the M1A or FAL isn't the way to go. LMT, KAC, Larue and FN SCAR Heavy if you need it to work.
