While in college, my Introduction to Hebrew Bible professor (may Hashem extend his days and fill his life with nachas) stated that there were three beliefs regarding the placement of the Temple:
according to the Masoretic text of the Torah, wherein the Temple was to be built on only one place, that place being the area now known as the Temple Mount in Jerusalem;
according to the Samaritan text of the Torah, wherein the Temple was to be built on only one place, that place being Mt. Gerizim (where Samaritans continue to practice their rituals);
anywhere (of which the ruins of the Temple at Elephantine in Egypt are evidence).
He said that the competing claims of the Masoretic Jews and Samaritans arose from traditional rivalries between both movements. And Jerusalem was specifically chosen to legitimize and promote the House of David (which came from near Jerusalem). The theory of other potential areas for the Temple or a Temple being built was shot down to emphasize that there was only one legitimate kingdom (Judah), only one legitimate ruling house (the House of David), only one legitimate temple (in Jerusalem), and only one legitimate High Priest (at the Temple in Jerusalem, near the origin of the House of David, in the Kingdom of Judah). In short: the consolidation of the belief, of there being only one possible or divinely-appointed place for the Temple, was political in origin.
This having been said, what would prevent Jews from building a temple today (a real one, according to the instructions of the Bible, not the Temple-replicas that synagogues are supposed to be) on a place other than the Temple Mount?
From what I remember of hebrew school, a large temple like those once on the Temple Mount would require animal sacrifices, something most modern Jews would probably not appreciate. I went to a reform synagogue and I never heard anyone suggest building a temple.
Except that David lived well before there were Samaritans in the area. The Samaritans arrived only after the Ten Tribes were exiled, about 300+ years after David lived.
Except that Jewish tradition teaches that the Temple was actually in the tribal portion of Benjamin, not Judah (they both shared the city of Jerusalem).
Very simply, the fact that Jewish law forbids it. The Temple can only be built on the spot where it existed. The Altar has to be built on the exact same spot that it was previously built on. One cannot have the Temple built anywhere else.
Well, since the official position of Reform Judaism is that there will be no messiah, no restoration of the Temple or an ingathering of the exiles, I suppose you probably would not have heard of it.
I think that more important than having the Holy of Holies in the precise spot is having the sacrificial altar in the precise spot.
Not necessarily. Maimonides says that “if someone (who meets certain other messianic criteria) manages to rebuild the Temple (and gather in the Exiles, and some other things), then he is definitely the Messiah,” which sounds like the building may be done before a person is considered to be the Messiah.