So I'm not sure I quite follow the timeline of the Amulet of Kings being the Stone for White-Gold. I thought the Amulet was given to Alessia during the slave rebellion against the Ayleids, so that seems to indicate that it is not as old as the Tower it is supposed to be bound to, since the Ayleids were already in possession of the Tower.
Additionally, if the Stone for White-Gold is the AoK, then the Tower should have been deactivated before the Umbriel crisis. If so, then why was WG Tower to be used to free the City of Vile's circumscription?
Finally, if the answer to the above to discrepencies is that a new Stone was substituted for an old Stone (AoK for whatever the Ayleids used; some new Stone for the shattered AoK), then what is the fuss about the Thalmor deactivating Towers. So long as the Tower itself is still in tact, couldn't a new Stone be created to bring it back online.
One final point, is the notion of Stone substitution what happened with the Mantella and the Heart of Lorkhan for Walking Brass? If so, it could serve as a proof of concept that Towers can be re-activated with an appropriate replacement Stone.