The professors debunk the Ethopia legend on the grounds that Aksum (the city where the Ark is located) did not exist at the time of Solomons reign in Israel--a valid objection.
However Hancock's thesis is that the Ark was removed from Israel during the reign of Manassa (about 650 BC--long after the time of the Menaleik legend which would have been in the middle of the tenth century BC) and taken to a newly constructed Temple on Elephantine Island in the Nile in Egypt where it was located for about two hundred years. There is significant historical support for the existence of the Temple; there is a large body of correspondence which survives between the priests of that temple who were Levites and the priests in Jerusalem and later with the excile community in Babylon.
Hancock trances the Ethiopian history on the ground from Elephantine to an island in Lake Tana where the Ark reposed for several hundred years when it was then moved to Aksum.
Supporting evidence is found in the history of the Knights Templer who were organized during the Crusades in Jerusalem for the purposes of finding the Ark. After substantial investigation there, they moved their focus to Ethopia and there is a body of evidence placing red haired Europeans in Ethopia in custody of the Ark just before King Philip destroyed the Templers.
Hancock's analysis hangs together very well--he is not part of the professional education community so they reject his work however I see the book as an excellant analysis of the evidence and it makes a good case that the object in the church of Christ Mary in Aksum is in fact the Ark.
I don't think there is any evidence in support of the "buried under the Temple Mount" argument--I think if it were buried under the Temple Mount or for that matter anywhere in proximity to Jerusalem, it would have appeared when Ezra reconstructed the Temple.
It should be noted in our time that Issah Chapter 18 is a prophecy that in a time of famine and tribulation in Ethopia, the Ethopians will send a gift to the mount of Zion which is an ensign (which the Ark was)--return of the Ark to Jerusalem would be likely to initiate a resumption of the sacrificial worship system, with or without reconstruction of the Temple; thus making possible fulfillment of the prophecy of Daniel 9:27 that the sacrifices would be interrupted at the commencement of the Great Tribulation in Matthew 24:15.