Whatever happened there, and happens now, is a matter of intense concern.
It is not a stretch to conclude that it is the prime reason, or at least among the prime reasons, why Jews and Palestinians have not moved beyond where we are to a final settlement of our disputes.
There is no decent doubt that it was the site of the First and Second Temples, with the present structure of the Mount created by Herod in order to hold the Temple that he created on the basis of what the returnees from Babylon built on the site of what had been destroyed.
The Western Wall is a retaining structure, meant to firm up the Mount and whatever is built upon it. Since the destruction by the Romans, it is as close as Jews have been able to get to their holy site, and so has become the most holy place for religious Jews, as well as a place to visit by Jews who are pondering their religiosity.
Muslims assert that Jews had only a brief period in Jerusalem in years past, and have no rights to its holy places that approaches in legitimacy their claim. In their stories, Muhammad rose to heaven while riding his iconic horse from the mountain, perhaps from the rock under the Dome of the Rock (also said to be the rock on which Abraham meant to sacrifice Isaac), or perhaps from the al Aqsa Mosque, a long stone throw to the south.
Muslims say that the site is also holy to Christians, but that is as much a problem as their other claims. It appears along with the statement that Muslims are the proper guardians of the Church of the Holy Sepulchre, an assertion that fits oddly with the record of Muslims having hounded Christians from Bethlehem, Ramallah, and Nazareth, where Christian majorities have become shrinking minorities under pressure.
What the Crusaders did have on the Temple Mount during the years they controlled it was a stable for their horses. In recent years the Muslims have greatly expanded that area, at a level below ground, to accommodate more people during times of prayer. Along the way, they have produced tons of rubble including whatever might have been found from the First and Second Temples by archaeologists. Israel's Antiquities Authority has protested, time and again, but government ministers have avoided the enforcement of the country's laws against despoiling ancient sites, seemingly to avoid yet another upsurge in Muslim violence.
The most crucial decision that pretty much created the present was taken by Moshe Dayan in his role as Defense Minister in the immediate aftermath of the 1967 war. He ordered the removal of the Israeli flag that soldiers had put above the Dome of the Rock, and conceded control of the Mount to Muslim religious authorities.
There are issues among Muslims as well as between Muslims and Jews. The family of the Jordanian Monarch claims an ancient right as guardian of Muslim holy places in Jerusalem, and the peace treaty between Israel and Jordan recognizes that status.
"Israel respects the present special role of the Hashemite Kingdom of Jordan in Muslim Holy shrines in Jerusalem. When negotiations on the permanent status will take place, Israel will give high priority to the Jordanian historic role in these shrines."
Palestinians claim their own priority, producing occasional frictions when it is time to appoint a new head of the religious authority, or when--as at present--Jordan asserts its concern for the Temple Mount in a period of tension, and the Palestinians assert their own priority.
Jordan, as well as a number of other Arab countries have recently cooperated with Israel on opposition to the deal with Iran, and opposition to the Islamic State. However, they also have been shrill in criticizing what they call Israeli provocations on the Temple Mount. The Economist put it this way
"Jordan (and the ) Gulf states also joined in, with angry condemnation from the Saudi foreign minister and the Kuwaiti cabinet, among others. They may all long for a more open relationship with Israel, but public opinion will keep their courtship in the closet."
Jews are allowed to visit the Temple Mount, except during times of prayer (Friday and Muslim religious holidays), but are not allowed to pray. The latter provision, like overlooking destruction of archaeological artifacts, is meant to avoid provoking Muslims' assertions of priority.
Orthodox rabbis have added to restrictions against Jews. The Rabbinate is responsible for a sign warning Jews to stay away from the place altogether, out of fear that they might tread upon the location of the Holy of Holies. There are competing assessments of just where it was. According to tradition, it was the resting place of the Ark in the First Temple. During the period of the Second Temple, entry was limited only to the High Priest on Yom Kippur.
There are inspired Jews who wish to pray on the Temple Mount. Some have been preparing ritual garments and instruments, searching for a red heifer whose ashes will allow a ritual of purification, and demanding the removal of Muslim buildings so the Third Temple may be constructed and Judaism returned to the rites of animal sacrifices as described in the Torah.
In years past, Jews were allowed to wander throughout the platform of the Temple Mount (an area of about 100 meters by 200 meters), and to enter the Dome of the Rock and al Aqsa. For some time now entrance to the Dome of the Rock and al Aqsa has been denied to non-Muslims, and the total closure of the Temple Mount has occurred when police expect trouble. There has also been an increase in the activities of Muslim extremist organizations, whose members see their mission as cursing, spitting, and otherwise disturbing Jews who dare visit their holy site.
We can summarize Israel's official posture on the Temple Mount as recognizing its political tinder, and seeking to avoid trouble while also assuring at least a minimum of rights for Jews, and keeping open the issue of sovereignty. The police have responded to violence with massed force and arrests. When times are tense, the police also forbid the entry to the Temple Mount to young Muslim men (those under the ages of 40 or even 50, depending on circumstances). This can produce confrontations outside the entrances to the Temple Mount or outside the walls of the Old City, with some violence and arrests.
Like other elements of the Israel-Palestine (or Israel-Arab) confrontation, the issues associated with the Temple Mount are mired in competing narratives, with an intensity of belief clouding any effort to agree about historical realities. While Muslim religious authorities control what usually happens on the site, Israel has shown a capacity to intervene with force and to overlook international criticism from Muslims and others. Neither side is willing to concede to the other more complete legitimacy and control.
We can guess that after the last of the refugees from 1948 has left the scene, and perhaps even if UNWRA ceases to care for their descendants, intensities associated with the Temple Mount/Noble Sanctuary/Haram al Sharif will remain to foil any effort to settle our conflict.