Solomon's Temple was NOT in the City of David.

If you think it was, I suggest you read 1 Kings 8:1, the account of The Ark being placed in The Temple.

"Then Solomon assembled the elders of Israel, and all the heads of the tribes, the chief of the fathers of the children of Israel, unto king Solomon in Jerusalem, that they might bring up the ark of the covenant of Yahuah out of the city of David, which is Zion."

And also 2 Chronicles 5:2

"Then Solomon assembled the elders of Israel, and all the heads of the tribes, the chief of the fathers of the children of Israel, unto Jerusalem, to bring up the ark of the covenant of Yahuah out of the city of David, which is Zion."

David's Tabernacle was not on the same land as Solomon's Temple, that's where these "The Temple wasn't on The Temple Mount" people are confused.

The land that Solomon built The Temple on, David purchased after the whole Census and Plague episode recorded in 2 Samuel 24 and 1 Chronicles 21, (and referenced again in 2 Chronicles 3:1).  So it can't possibly be the same spot near the Gihon Spring where David placed The Ark when he began his reign in Jerusalem.

In 2 Samuel 24:23, though English translations sometimes obscure this, Araunah (Ornan in Chronicles) is called a King.  He was the King of a separate Kingdom, so the land he owned was clearly not within The City of David.

There are also many reasons why you wouldn't put a threshing floor anywhere near a spring.

Now I don't know if the Second Temple was on the same spot as the first.  But I believe the evidence places the Second Temple's Holy of Holies about where the Al-Kas Fountain currently is, and that the Antonia Fortress was where the Dome of The Rock is.

Prophetic verses get used to back up saying there is a spring or river under The Temple.  Ezekiel 40-48's Temple will be no where near the same spot as Solomon's, it's miles north of Jerusalem, I've argued about where Beth-El was. And by then events like the 7th Bowl of Wrath will have totally changed the geography of the region.

In both Poetic and Prophetic books Zion gets used poetically, it doesn't always mean it's strict Geographical definition it has in the two verses that started this post.