There is no need for a physical 'Third Temple', JESUS is the Third Temple! John 2:19-22 reads:
Jesus answered them, "Destroy this temple, and I will raise it again in three days." They [the Jews] replied, “It has taken forty-six years to build this temple, and you are going to raise it in three days?” But the temple he had spoken of was his body. After he was raised from the dead, his disciples recalled what he had said. Then they believed the scripture and the words that Jesus had spoken.
IF a physical 'Third Temple' were to be built on the SAME site as Solomon's Temple and Herod's Temple, it would most likely be on the Ophel mound over the area of the Gihon Spring, in the old City of David to the immediate southeast of the traditional Temple Mount precinct. THAT would be less 'upsetting' to the mohamedans. As many Israeli Jews are secular (41%), the location of a 'Third Temple' on this supposedly 'incorrect' site, would not make its construction as difficult as building on the incorrectly named 'Temple Mount'.
OH boy now you done it. I had that very same argument on here a few months ago.
You are correct-the Temple was built in the city of David and NOT on mount moriah.
Like you stated-the site was near Mount Moriah and had to have running water to service it.
Being a sometime resident of Jerusalem, I tend to disagree with the theory that the Ophel mound was the location for the Temple, simply because the area is too small.
The spring itself is also lower than where the Temple floor would be, making transport of water rather difficult, depending on the technology used.
I believe there are/were recent excavations in the area to see if a structure existed; I do not know the results.
Anything is possible, especially with respect to the precise location of the original Temple (and new Temple).
It’s important to not be dogmatic about that since this is still playing itself out.
That said, it will be built.