I looked into history (am a huge nerd) of the Roman empire and it appears to me that the claim of Rome as the legitimate successor of Peter back in the days were politically charged. Rome lost all power and authority as the capital of the Roman Empire once Emperor Constantine officially moved the Seat to Constantinople (initially named Nova Roma, later changed to Constantino-polis, or City of Constantine).

Rome lost all power and claims at that point, and the Barbarians only hastened the fall of that city.

Fast forward a few hundred years, the Barbarians have now converted to Christianity and the political class (mostly powerful merchant families including the Venetians) obviously must do something to regain "the glory of Rome" .

Enters Charlemagne and his genius conquest. The political class at Rome immediately seizes the opportunity and crown him the Holy Roman Emperor. Mind you, the real and legitimate Roman empire is still existing in the East at Constantinople uninterrupted since its official relocation as mentioned previously.

This ultimately led to rising tension over the next few hundred years which finally resulted in the sack of Constantinople by the Venetian scums and their BLM puppets disguised as crusaders.