Here’s what I’ve determined:
The place was a synagogue which has been reverenced since the 4th century the site of Pentecost and the Last Supper. In the 13th century, Crusaders were surprised to find a tomb at the site. THE TOMB WAS NOT DIRECTLY BELOW THE “UPPER ROOM” (”Cenacle”); it was underneath the enlarged site used to build a larger church. So, this dispels doubts about a Pharisee building him home on a tomb site. For whatever reason, the Crusaders believed the tomb to be that of King David, and constructed a sarcophagus for it. In doing so, however, they made impossible for modern researchers to validate their presumption.
The lack of obvious identifiers is consistent with ancient historians: Josephus relates that Herod tried to loot the tomb, but found the tomb already looted; it’s not even certain that there are any remains in the tomb.
Whatever agenda the Crusaders may have had, the local Jewish population also came to believe that it was the tomb of David (although the larger Jewish community largely rejects the notion). Part of the confusion stems from the bible itself, which seems to purport in one place that David WAS buried on Mount Zion (the City of David) [1 kings 2:10], but also that he was buried with his ancestors [”slept with his fathers”], leading some to believe the passage refers to Bethlehem. But I’m not sure “slept with his fathers” MEANS he was buried at the site they were buried at.
Forgot this tidbit:
Of course, the New Testament ALSO refers to BETHLEHEM, rather than MOUNT ZION, as the city of David. But this plainly is a separate identification than the City of David referred to as David’s capital.