Polycarp was known as the oppose of heretics.
Valentinus came to Rome in the time of Hyginus, flourished under Pius, and remained until Anicetus. Cerdon, too…Marcion, then, succeeding him, flourished under Anicetus. But Polycarp also was not only instructed by apostles, and conversed with many who had seen Christ, but was also, by apostles in Asia, appointed bishop of the Church in Smyrna…always taught the things which he had learned from the apostles, and which the Church has handed down, and which alone are true. To these things all the Asiatic Churches testify, as do also those men who have succeeded Polycarp down to the present time -- a man who was of much greater weight, and a more stedfast witness of truth, than Valentinus, and Marcion, and the rest of the heretics. He it was who, coming to Rome in the time of Anicetus caused many to turn away from the aforesaid heretics to the Church of God, proclaiming that he had received this one and sole truth from the apostles... John, the disciple of the Lord…exclaiming, "Let us fly, lest even the bath-house fall down, because Cerinthus, the enemy of the truth, is within." And Polycarp himself replied to Marcion, who met him on one occasion, and said, "Dost thou know me?" "I do know thee, the first-born of Satan" (Irenaeus. Adversus Haeres. Book III, Chapter 4, Verse 3 and Chapter 3, Verse 4).
Valentinus, Cerinthus, and Marcion are considered by Catholics and others to have been Gnostic heretics, while Hyginus, Pius, and Anicetus were claimed bishops of Rome.
These quotes show that the supposed Roman bishops did not have a higher leadership role than Polycarp of Smyrna had. It took the stature of the visiting Polycarp, Bishop of Smyrna, to turn Romans away from the Gnostic heretics who flourished under wishy washy leadership from the named bishops pf Rome.