funny you should mention: “the cornerstone Herod set when he built the ‘Temple Mount’”; I had my hands all over it back in 2014. The Israelis have been excavating all under the Temple Mount area as part of the “City of David” Excavation. You can google it to get the info. It is awesome to stand on the very point that the entire Temple Mount rests on. In fact, I have walked the entire length of the Western Wall underground - all the way from the City of David to the exit at the Kotel Katan in the Arab Market, near the Damascus Gate. The Israelis are starting to allow visitors to most of it and it is well worth the tour if you ever visit Jerusalem. BTW, I have seen the “religious based” Excavations come and go. I am somewhat leery of them. I worked with Vendyl Jones and was a good friend for a lot of years. His financing was from Christian Zionists, and there is nothing wrong with that. However, most of the volunteers came with the idea that they were ushering in the Messiah and were on a mission. If that is why they came, they were disappointed. Unfortunately, I have run across a few Charlatans who use the allure of finding “Temple Treasure” to get funding for themselves. I loved one of Vendyl’s responses when people asked him about “when will Jesus Return?” His response was always I don’t wait for Messiah to come, I work for Messiah to come. He suggested that they not look up, but just keep working.
When my son went to Israel he went to work in the settlements... not to evangelize, just to work... That is the proper way IMHO. The locals say they 'can see Messiah' in folk like him.