Actually, Jerusalem and the city of David are not identical in the Bible. Mount Zion is the city of David (cf. 2 Sam. 5:7). Jerusalem was originally inhabited by the Jebusites (Jos. 15:8), and encompassed a much larger area around and including Mount Zion (cf. 2 Kings 14:20). Jerusalem is never called the city of David.
Of course the undeniable fact is that in the NT we see the ultimate expansion of Mount Zion/Jerusalem to include all the nations of the earth. The conquest of Jebusite Jerusalem by the nation of Israel is typical of the spiritual reality we see happening today.
The NT interprets the Old, not vice versa. (I noticed you really have little to say about the NT to make your position.)
The earthly is typical of the heavenly reality. Jerusalem above is free. You just have not gotten there yet. But it’s only a matter of time.
"Nevertheless David took the stronghold of Zion; the same is the city of David
And this from Wikipedia:
"The Ophel, העופל, perhaps meaning "fortified hill," or City of David is the name of the narrow promontory beyond the southern edge of Jerusalem's Temple Mount and Old City, with the Tyropoeon Valley (valley of the cheesemakers) on its west, the Hinnom valley to the south, and the Kidron Valley on the east. The previously deep valley (the Tyropoeon) separating the Ophel from what is now referred to as the Old City of Jerusalem currently lies hidden beneath the debris of centuries. Despite the name, the Old City of Jerusalem dates from a much later time than the settlement in the Ophel, which is generally considered to have been the original Jerusalem. The Ophel lies outside the current Old City walls, but it was once surrounded by a city wall. This wall was discovered by the engineers of the Palestine Exploration Fund at the south-eastern angle of the temple area, 4 feet below the present surface level. Since the Books of Samuel credit David as the first Israelite ruler of the city on Ophel, the archaeological remains of the city are usually referred to as the City of David. The Ophel was considered part of Jerusalem until the 12th century CE, but after that point became regarded as a separate village.