I don't care for the politics or accessbility, objects derive part of their value from the place they came from and should be kept in their historical context as much as possible.
The Elgin marbles should be in the Acropolis Museum, Agamemnon's mask should leave Athens for Mycenae. The horses of St. Mark's can stay in Venice, because they are part of Venetian identity much more than they (likely) were in Constantinople. It's not perfect by any means, but the political sensitivities of today (either way) don't make much sense to me either.