Today we celebrate the feast of the Chair of St. Peter which is different from the feast of St. Peter the man which we celebrate on June 29. If we recall that the king's chair is the only chair in a throne room, and if we recall how the officer commands, "Let all be upstanding," allowing the judge to be the only one taking his chair, then we see that a chair is a symbol of authority.
In the first reading Peter told the presbyters to lead through good example rather than by issuing commands. We can deduce that he practiced what he preached, going easy on making direct demands of people.
The Gospel tells us that Peter, and presumably his successors, had an unique authority symbolized by Jesus giving him the keys of heaven. The other part of that commissioning was not restricted to Peter alone. Two chapters further on Matthew records how Jesus, speaking to all the disciples said what ever they bound on earth would be bound in heaven; whatever they loosed on earth would be loosed in heaven.
In the year 250 St. Cyprian in North Africa argued with the pope over the matter of the pope's installing bishops in Africa. Cyprian thought such matters should be handled locally. However, he wrote a clear defense of Rome's position, saying that no church that was not in union with Rome had the right to call itself a Christian Church.