If I remember correctly some of the tribes (Vandals/Goths) were Arians, and also really powerful. With Byzantium pretending to still be in control and delegating themselves out of existence by recognizing Germanic overlords as governors, who was in practicality protecting the Church or the pope? I know papal authority took a few more centuries to really solidify, but popes are important folks in this time period too. Just trying to wrap my head around this kind if fallow period where my knowledge of history drops off.
Sorry if this is confusing - I'm on mobile and trying my best to communicate my question.
I can give you a short answer (can't seem to find my sources, unfortunately):
Immediately following the fall of the Western Empire, the Ostrogoths became the "protector" of the papacy. They were Arian, but felt that the Roman legacy of the papacy was respected enough to survive. Some papal elections were influenced by them (and the Eastern Empire), but the Ostrogoths otherwise largely stayed out of Catholic religious affairs.
Rome was returned to imperial control as part of Justinian's campaigns to return the Empire to its original extent. Eastern protection was strong at first, but tapered off as the ability of the Eastern emperors to hold onto the territory they'd reclaimed dwindled. Eventually the Eastern emperors held only a few cities, including Rome, due to the Lombard invasion pushing them out.
The Lombards eventually took Ravenna, an important city controlled by the Eastern emperors, and looked like they'd threaten Rome. The pope decided to head north and forge an alliance with the Franks, who in turn invaded the Lombard kingdom several times.
Charlemagne himself defeated a Lombard attempt to conquer Rome and added the city to his growing empire.
Hopefully somebody with better source access than I has a few to back up the general narrative here.