I mean, they had all the Islamic Holy Cities and such. Why didn't they declare Caliphate status? Was it because they weren't descendant from Mohammed? If that was the case, couldn't they simply have faked their relation to him?
They did declare a Caliphate. One of the titles of Ottoman Sultans was Caliph of Islam. The first to do so was Murad I in the 14th century when he conquered Adrianople and renamed it Edirne, establishing it as his capital. The title was used all the way until Atatürk's reforms, when it was abolished.