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.
Not sure really. Mostly what I find online is focused on the question of King David rather than the tomb itself, if such a thing is even there. At http://www.jewishvirtuallibrary.org/jsource/Archaeology/Davids_Tomb.html there is a mention of a Benjamin of Tudela who, in 1173, tells of the miraculous finding of the tomb during renovations to a church there. This would seem to suggest that something like a tomb, at least, was uncovered at that time, though it may be impossible to actually physically confirm this today. Did these people have the knowledge to be able to identify such a Jewish tomb, or did they perhaps just find a hole and give into wishful thinking? Or were there perhaps bones or an ossuary of some kind?
Benjamin is also mentioned at http://www.jpost.com/Video-Articles/Video/CITYsights-A-tale-of-two-tombs which goes further and explicitly says that this man's writings were the "first evidence that the site was viewed as David’s burial place." It would appear, then, that at the minimum there is no ancient witness that this location is the tomb of David at all, which certainly makes this whole episode a little dubious. At least to me.
On the other hand, whatever evidence WAS there, must have been fairly convincing, since it convinced the local Jews of the day, despite the antagonism with the Crusaders and the theological significance of the Last Supper altar being so close to King David.