SK, yes if you were incredibly unlucky and somehow ended up with a smooth carrier the rifle still should function properly except that the forward assist couldn't grab the carrier and thus wouldn't function. That would be the only feature that wouldn't work.
I'm a milspec guy and like the full auto/M16 quality BCGs from BCM, Daniel Defense, Colt, PSA and a few others. I have two from BCM, one from DD and never an issue with any of them. You'll find a lot of people who say it doesn't matter whether the milspec steels and other specs are adhered to but I say why not go with what the military demands, they do know a thing or two about the M16/M4 after 50 years.
Anyway all that may be moot now. The commercial BCGs should work and it's hard to find anything so you may have to jump on something, anything if you're desperate. I don't know how backorders are handled by the various vendors but hopefully someone else who has bought lately can chime in. The only thing I've seen is some of the boutique BCGs like Black Rain, FailZero and Young tend to be available since they cost about $250+. Good luck!