You've got to be kidding me !
Abu Bakar (RA) consolidated Islam & stopped all of them False Prophets from tearing Islam apart !
Omar (RA) took it a step further by Institutionalizing many Islamic Paradigms & was instrumental in increasing the Islamic Empire with many a Non-Muslims, including the Patriarch of Jerusalem, admiring Omar for his fairness & simplicity in stark contrast to the decadence & corruption of their previous Masters. And he was assassinated by that vermin Firoze because the Persian couldn't come to terms to how his Civilization was defeated by a bunch of bedouin out of Arabia...it had nothing to do with his treatment or mistreatment of new subjects.
Furthermore Usman (RA) was instrumental in institutionalizing Islamic Financial Paradigms & giving the Islamic Empire a strong footing to be a Islamic Welfare State. His only shortcoming was that he wasn't as strong of a personality as his two predecessors & couldn't stomp his authority as needed.
And no the Prophet (PBUH) did not appoint Ali (RA) as his successor or anything like it. Nor did he (Ali) clean up any mess but created a hell of a situation that saw the first Civil War of Islam due to his refusal to punish the assassins of Usman (RA).
I would've taken Ayesha (RA) as the Caliph in his stead but I dunno if she would have agreed to it or not !
But there was many Sahaba still left so I'm sure we could've found someone !