I was under the impression, as are many people I have met, that the Temple Institute believes we are allowed to rebuild the Beis Hamikdash ourselves (without moshiach).
Either they really do believe that, or they play up the misconception in their ads because they know if brings in the dough.
For those saying it was located elsewhere, those people are probably what you would call Biblical Nihilists. They base all of their theories and beliefs off of whatever is the opposite of the words of the Tanach, even if archeological evidence suggests otherwise. In secular society, as stupid as this approach is, it sells a lot of books. What I’ve always found ironic is that if you are purposely believing the opposite of what the Bible says, you’re actually STILL basing all of your beliefs on the Bible.