Since you have brought this up several times, Liseux, I think it is about time it was addressed.What happened already in the 1st century I posted earlier, with Clement of Rome writing to the Corinthians to discipline them for rebuking their overseers. He did settle the dispute, as they recognized his hierarchical position. (as a bishop)
I have read Clemen't letter to the Corinthians (written shortly after the death of Paul and Peter) several times. How privileged we are to have access to such an early letter written by such an early overseer in the Church, a man who is likely to have been the fellow-helper of the apostle Paul whom Paul mentions in Philippians 4:3! This letter does not seem to contain material interpolated by later writers such as the letters purported to have been those of Ignatius apparently have.
Clement didn't write the letter to "discipline them". The problem was not merely that the people of the Church at Corinth "rebuked" their overseers. They wanted to remove them from their overseership.
Clement appeals to them to consider what they are doing. He shows how God dealt with people in the past who rebelled against the leaders whom God had set. He asks them to take up that letter of the blessed Paul and reread it. When Paul wrote to the Corinthians, their partialities were for renowned men such as Peter or John or one of the other apostles. Now they were for the young upstarts who wanted to bring down the overseers and become overseers themselves.
I don't see this letter as showing that Clement had some special authority as an early "pope".
The Corinthian church did not repent when Paul rebuked them for their divisions and partiality. Rather, by the time Clement wrote to them, they had gone from bad to worse.
I don't know how they responded to Clement. Do you have any historical evidence that they repented, and supported the existing overseers? If so, I would be interested in this evidence as well as your source.