it's been mentioned in numerous interviews that each member of the band recorded their vocals separately.
all the videos of them singing in the studio aren't any vocal takes that ended up on the album. they were all done for promotional purposes. AKA, they were faked.
If wondered if that wasn't the case, and it wouldn't surprise me if it were so. I could be wrong, though, but I think I've read statements in interviews that indicate that at least some of the vocals were recorded live in the studio. There has been a lot of contradictory information about the album, in any case.
And it still raises the question as to whether Dave is doing any singing. Let's say that all of the videos of them singing are faked--we still see Brian giving David a part to sing. Could be totally phony ('let's try to make Dave look more involved here'), but it could also be that Dave isn't credited, or that he did sing in the studio, but Brian decided not to use his parts.