“while those debating you have cited at least one early church father whose account contradicts that supposed history”
If you are referring to Pope Gregory never denied his Papacy? That is a matter left for interpretation, and interpreted by Non Catholics as a crutch to debunk Papal Succession...
Irenaeus: "The blessed apostles, then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate. Of this Linus, Paul makes mention in the Epistles to Timothy. To him succeeded Anacletus; and after him, in the third place from the apostles, Clement was allotted the bishopric. . . . . To this Clement there succeeded Evaristus. Alexander followed Evaristus; then, sixth from the apostles, Sixtus, was appointed; after him, Telephorus, who was gloriously martyred; then Hyginus; after him, Pius; then after him, Anicetus. Sorer having succeeded Anicetus, Eleutherius does now, in the twelfth place from the apostles, hold the inheritance of the episcopate. In this order, and by this succession, the ecclesiastical tradition from the apostles, and the preaching of the truth, have come down to us. And this is most abundant proof that there is one and the same vivifying faith, which has been preserved in the Church from the apostles until now, and handed down in truth (SOURCE: Iraeneus Against Heresies, Volume I, Book III, Para 3)