While I think that there is certainly to be made that the M14A1 was present in Vietnam in some numbers, for at least a short period of the war, I just don't think that it's crying out to be included as a weapon in the game.
Automatic variants of the M14 were, by virtually all accounts, mediocre weapons. Even firing prone, with a bipod, the M14 has poor accuracy firing bursts, on account of having very high recoil. Like the BAR, it only had a 20-round clip, feeding from under the gun, which severely limited its usefulness as a fire-support weapon.
The M14A1 was not substantially changed from the vanilla M14; there were no major changes to accommodate its use as a LMG (a heavy-barrel version, the M15, was actually rejected in favor of the M14A1). The only differences were some modest changes to the stock and hand-guard, and the addition of a bipod. There's nothing to suggest that the M14A1 did any better in automatic fire than the standard version.
Furthermore, the M14A1 was intended as a direct replacement for the BAR, and in terms of function, is very similar - an automatic rifle, firing a full-power cartridge from a 20-round magazine.
Gameplay-wise, I don't think that there is a strong case for adding such a lackluster and unremarkable weapon to the game, which would more-or-less replicate a less-distinctive BAR on the US Army side. Especially when there are a number of other interesting Southern weapons, such as the Stoner 63 Rifle / Machine Gun, or the Carl Gustav m/45 (a.k.a S&W M76) that have yet to be added to the game.