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.
Among the large amount of evidence that goes against the ‘Temple Mount’ being the site of Herod’s Temple, are ROMAN coins dated
to 20 years AFTER Herod’s construction, found at a dig at the ‘wailing Wall’.
The only source of flowing water in the Jerusalem of Jesus’day was the Gihon Spring in the Old City. Jews were required to perform ‘mikvah’ using flowing water. The Temple Mount is located some distance away from the Spring and at a higher elevation! It makes absolutely no sense that they would bathe in the Spring, or the nearby Pool of Siloam, then walk uphill on a dusty road to the Temple. That would increase the risk of being made ‘unclean’.
There are so many issues with the Temple being located on the Temple Mount, that it is nigh impossible for it to have been there.