Its hard to speak on what peter did or didn't say on the matter because so few of his words are recorded after this point. There are two possible sources of clues. The Acts of the Apostles and writings of early Christians.

In Acts 15 we see the Apostles holding the first council on the topic of circumcision. There is much dispute until Peter stands up to speak. He speaks with authority and the bishops fall silent and listen. This is the authority which we believe that the pope was given. He can't say anything he wants. He is not a king. But when there is a dispute among the bishops his words carry authority and strongly influence the bishops. The decision is then made by all the bishops.

There is some evidence among the letters of the early Christians that Peters office held other responsibilities such as coordinating the bishops and disseminating writings to the other bishops.

I don't have the time to go through all the writings of the church fathers with you, but I will point out this one: https://www.newadvent.org/fathers/0103303.htm

It was written around 180 and lists the succession of specifically peters office on account of its preeminent authority.