If the Palestinian Arabs assume responsibility over the site they will have freedom to destroy much more.

Journalist Ilan Ben Zion reported in December 2012: "The Muslim authority managing the Temple Mount on Sunday dumped tons of unexamined earth and stones excavated from the holy site into a municipal dump, in violation of a High Court injunction, Maariv (Hebrew Daily, ed.) reported on Monday." Israel’s top court in September 2004 prohibited removal of earth from the Temple Mount and ruled that, should it be necessary [to do so], the Antiquities Authority must be notified a month in advance so it may examine the earth for artifacts."

"Jews regard the Temple Mount as their holiest site, where the First and Second Temple were located. Muslims call it the Noble Sanctuary and regard it as their third holiest site after Mecca and Medina. According to the existing arrangement, the Jerusalem Islamic Waqf, or trust, administers the Temple Mount complex. Despite the High Court of Justice’s ruling, the Waqf has reportedly removed large piles of dirt from the Temple Mount in recent years and dumped them in the valley east of the Old City walls, provoking an outcry from biblical archaeologists and Jewish groups.

"Tzachi Dvira, the archaeologist managing the team that sifts through soil excavated from the Temple Mount, told Maariv that mounds of earth containing historic relics were carted off and dumped on Sunday without notification and before archaeologists could investigate them... Soil from the Temple Mount that had been removed by the Waqf to the Kidron Valley in recent years has yielded “tens of thousands of finds, including signet rings from the First Temple era, painted floor tiles from the Second Temple era, ancient gold coins, and horseshoe nails and arrowheads belonging to the Knights Templar, who stabled their horses in Solomon’s Stables,” Dvira said.

Suzanne Singer, a contributing editor to the Biblical Archeological Review reported in September 2000 “Large-scale illegal construction on the Temple Mount and wholesale dumping of earth in the nearby Kidron Valley resumed this spring…” 

....

On Jan 6, 2018 Judith Abramson reported in Jerusalemonline "Hebrew University archaeology doctoral student Haggai Cohen Klonymus described to the Israel News Company (formerly Channel 2 TV) how Palestinian tractors and bulldozers arrived at an archaeological site where the ancient city of Archelaus once stood. The Palestinians completely leveled the compound in order to locate hidden archaeological treasures to sell in the antiquities market.  "Just as ISIS destroyed sites in Iran and Syria that were thousands of years old, the same situation is occurring here," he said. "This is a deliberate and systematic destruction of an archaeological site....It's just a tragedy."

In 2000 the Palestinians destroyed Joseph's tomb. Sidney Brounstein wrote for the Los Angeles Times "Oct. 8:  Where is the outrage? Imagine what would have happened if Jewish police stood by and allowed a Jewish mob to destroy a Muslim holy place!  Does the destruction of a Jewish holy place by an Arab mob while Palestinian police stand by (after promising to protect it) deserve no more than inclusion in a list of other damage done by rioters? Is this an acceptance of attacks on Jews and things Jewish as a normal part of life?" "It makes a mockery of any thought of giving Arabs any control of Jewish holy places. The destruction of dozens of such places in the Old City of Jerusalem from 1948 to 1967, along with the exclusion of Jews entirely from their most holy site, the Western Wall, was clearly of a piece with the current destruction."