Did Ali ibn Abi Talib, may Allah be pleased with him, believe that the Caliphate was restricted to the Ahl al-Bayt (the Prophet's household), which delayed his pledge of allegiance to Abu Bakr al-Siddiq, may Allah be pleased with him?

Ali, may Allah be pleased with him, did not see the caliphate as an obligation belonging to the family of the Prophet. His delay in pledging allegiance to Abu Bakr was not due to this, but rather to his belief in the necessity of consulting him on the matter. Indeed, Ali first pledged allegiance to Abu Bakr along with the people, and then renewed his pledge later after the death of Fatimah, may Allah be pleased with her, to remove any misunderstanding, and not because he was displeased with Abu Bakr's caliphate.