The early possessors of the Temple Mount were Cannanites – and their descendants still survive.
David took Jerusalem from the Jebusites ca 1000 BCE. Solomon built the Temple about 953 BCE. After Solomon’s time, the Jewish state split into two kingdoms, Israel in the north with its capital at Samaria, and Judah in the south, with its capital at Jerusalem. The Assyrians took Samaria about 750 BCE and marched the inhabitants off into captivity (these are the "Lost Tribes of Israel.)
The Babylonians took Jerusalem about 586 BCE. They destroyed the Temple, took the Ark of the Covenant, and marched the people of Judah off into captivity (this is the “Babylonian Captivity.”) About 60 years later, Cyrus the Great of Persia conquered Babylong and allowed the Jews to return to Jerusalem, under Zerubbabel, who built a second temple (about 515 BCE.) This temple was “so small and mean that those who had seen Solomon’s Temple wept.”
Zerubbabel’s Temple was razed by Herod the Great in 19 BCE. Herod built a new Temple, following the biblical description of Solomon’s Temple. It is Herod’s Temple where the Holy Family sacrificed.
In the 60s of the Current Era, there was a Jewish revolt. The Roman General Titus (later Emperor) stormed Jerusalem and destroyed Herod’s Temple in 70 CE.
There was a second revolt in 133 CE. The Emperor Hadrian expelled the Jews (note: It was the Romans, NOT the Muslims – who didn’t yet exist – who expelled the Jews). Hadrian build a Roman city, Aleia Capitolina, on the ruins of Jerusalem. He also build a temple to Jupiter.
Mount Moriah today is larger than it was in Jesus’t time – and, politically incorrect as it may be, the Western Wall (the Wailing Wall) may be the reason. Some athorities believe this wall (which is a retaining wall, not part of any temple) was built when Hadrian enlarged Mount Moriah fror his temple.
St. Helena had her son, Constentine, restore the name Jerusalem. He did not lift the provision against the Jews returning, however, and Jews were not allowed back until 632, when the Muslims took Jerusalem (note: It was the Muslims, not the Christians, who allowed the Jews to return.)
Muslims believe that Mohammed ascended to heaven from the Rock on Mount Moriah (the rock on which Abraham was supposed to sacrifice Issac.) The Dome of the Rock (the building with the gold-plated dome) was built on Mount Moriah about 690 CE.
When the Crusaders took Jerusalem, they drove the Jews and Muslims out of the city and massacred them under the walls. Jews were not allowed to return until the Mamlukes regained control of the area about 1260 CE (note: It was Catholics who drove out and massacred the Jews this time, it was Muslims who allowed them to return once more.)
The Dome of the Rock has been on Mount Moriah longer than Soloman’s Temple, Zerubbabel’s Temple, and Herod’s Temple combined. To destroy it would be a crime of great magnitude.