To be clear, you should read on in the very next paragraphs of Irenaeus’ Letter.
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3The blessed apostles [Peter and Paul], having founded and built up the church [of Rome], they handed over the office of the episcopate to Linus. Paul makes mention of this Linus in the epistle to Timothy [2 Tim. 4:21]. To him succeeded Anencletus, and after him, in the third place from the apostles, Clement was chosen for the episcopate. He had seen the blessed apostles and was acquainted with them. It might be said that he still heard the echoes of the preaching of the apostles and had their traditions before his eyes. And not only he, for there were many still remaining who had been instructed by the apostles. In the time of Clement, no small dissension having arisen among the brethren in Corinth, the Church in Rome sent a very strong letter to the Corinthians, exhorting them to peace and renewing their faith. . . To this Clement, Evaristus succeeded. . . and now, in the twelfth place after the apostles, the lot of the episcopate [of Rome] has fallen to Eleutherus. In this order, and by the teaching of the apostles handed down in the Church, the preaching of the truth has come down to us.
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4Polycarp 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, whom I also saw in my early youth, for he tarried [on earth] a very long time, and, when a very old man, gloriously and most nobly suffering martyrdom, departed this life, having 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” (*Against Heresies *3:3:1-4 [A.D. 189])…
Maybe it’s just me, but it certainly SOUNDS like Irenaeus provides what you claimed to be missing: “
the Catholic argument of Peter and his successors as head of the Church.”
First, Irenaeus writes, “With that church [of Rome], because of its superior origin, all the churches must agree, that is, all the faithful in the whole world”. If all churches must agree with Rome, then Rome is the head of the universal Church.
Next, Irenaeus lists the successors of Peter: Linus, Anencletus, Clement, and so on.
So, we have Rome as the Church which all others must follow, and the successors of Peter listed.
You stated the following: “This would have been the most important argument against heresy if Irenaeus knew about it.”
I believe Irenaeus has silenced you, and if this were a game of chess, now would be a good time to tip your king.