Evolution of the Byzantine Papacy?

So, the Byzantine Papacy is the period from 537-752 in which the Byzantine Empire controlled a central portion of Italy, Rome, and had a deciding hand in the nature of the Bishop of Rome in the West. During this period Rome once more became a cosmopolitan city as ideas and money from the East flowed into the city. Notably Greek was the language of the Papacy bureaucracy over the local Latin majority for much of this period.

How would the local church have been shaped if the Byzantine Papacy had continued largely uncontended? Would Latin eventually have risen over Greek? Would a split in the Church still have occured? As well as other matters both large and small, secular and theological, etc.
 
maybe the germanic kingdoms to the north and west develop their own, indigenous branch of Christianity, with superficial differences to the Roman one, with the head of each sect under the authority of the local king. Some may, depending on the realpolitik of the time, temporarily accept Roman authority over church matters as a way of obtaining an alliance, support etc.
 
maybe the germanic kingdoms to the north and west develop their own, indigenous branch of Christianity, with superficial differences to the Roman one, with the head of each sect under the authority of the local king. Some may, depending on the realpolitik of the time, temporarily accept Roman authority over church matters as a way of obtaining an alliance, support etc.

So 'The Church' remains largely under Byzantine Control and it encourages seperate Churches to form.
 
If I may look at it from a different angle. The period was when the Emperor appointed Popes for over 200 years, which I suppose gave the idea the legitimacy of time. So WI Charlemagne and his successors claimed the same or similar right to appoint Popes deriving from the fact that its traditional for Emperors to appoint Popes? How could that play out?
 
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