It would seem to me that any competent and pious leader could fulfill the role, and it’s not like Muhammad’s bloodline is divine, right? How did this belief arise, and more generally, what led to the Sunni/Shi’a split aside from politics? I’m asking about the theological history I guess. Thanks!
Well, a few things. First, the Shia don't believe that the successor to Muhammad must be a "descendant" of the prophet. Ali was a cousin of Muhammad and his son-in-law, but not his descendant. All of the Shiite Imams other than Ali are descendants of Ali and Muhammad.
Second, the answer to your question depends in large part on which Shia you're talking about. Ali had his own claim to the Caliphate which is respected even by Sunnis. He is recognized by them as one of the four Rashidun, the rightly guided Caliphs. But what Ali was or the extent of his claim varies significantly among historical Shia sects. Most of our information about many of these groups comes from Sunni heresiographers and so must be taken with a grain of salt, but some groups, for instance, regarded Ali as being perhaps a kind of manifestation of God Himself. That is absolutely not the orthodox position of the Twelver Shia today, but all Shia (as far as I'm aware) for example append the shahada, the Islamic statement of faith that there "There is no god but God. Muhammad is the messenger of God." with the line "And Ali is the vice-regent of God."
Third, this basically points to the special claim of the Shia as Alids, that is, "Shia" itself is an abbreviation for "Shiat Ali" the "Partisans of Ali." Both Sunnis and Shia, in fact, agree that "any competent and pious leader could fulfill the role" cannot fulfill the role. That was the position of the Kharijites. For example, both Sunnis and Shias hold that the "Hadith of the Twelve Successors," that Muhammad will have twelve successors and they will be from Quraysh (i.e. Muhammad's kinsmen) is sound. Where they disagree is on the place of the "house of the Prophet" itself, including Ali.
As for the Sunni/Shia split in general, I've written on the topic previously before, for example here, that may be of interest: https://www.reddit.com/r/AskHistorians/comments/4x6588/when_did_the_fundamental_differences_between/
I realize this response is a bit rambling, I'm happy to take follow-up questions as best as I can.