Doubtful Christian authenticity, IMHO, in that I doubt a house would have been built on a tomb — although, real estate is tight there, and things do get buried, so it is possible.
Yes, that would make sense, but the more doubtful attribution is not the cenacle, but the tomb. IIRC, the traditions about the upper room being in this location are very ancient indeed, but the legends about the tomb of King David only date to the crusades. More than likely it is a case of its holiness drawing in other latter accretions, perhaps begun by medieval Christians but then being accepted by Jews and the like. It makes the Jewish protest kind of funny since, most probably, they are upset about the desecration of a tomb that doesn't exist, and is only thought to exist there in the first place because of its important Christian associations.
Is the question whether there is any tomb at all, or merely who the tomb was for? I presumed that there must be SOME tomb if there’s even a question of whether it might be King David’s tomb.
But if it WERE KIng David’s tomb, that’d only make it a more awesome place of the last supper. (Well, not quite: infinity + 1 is not greater than infinity.)