Here is a primary source that will provide further illustration that the early church had no knowledge of any kind of “petrine primacy.”
The “Apostolic Constitutions” was unknown in the West until the Middle Ages. Daniel O’Connor (“Peter in Rome”) dates the earliest form of the document in the early third century (that’s in the 200’s for those who are counting). There are multiple versions of this document: Latin, Greek text, Syriac, Arabic, and Ethiopic versions. The Syriac is certainly the earliest version of which we have any record; the Latin is derived from a Greek original which also closely approximates the extant Syriac text.
In the Syriac version there is one passage which relates to this study. This reference to Peter’s being in Rome is derived from the “Acts of Peter” and deals with the activities of Simon Magus …
The Ethiopic version is more elaborate and specific … This version also contains a list of those ordained by the apostles: “First in Jerusalem, James …. And in Antioch, first, Evodius [ordained] by Peter; and after him Ignatius, by Paul …. And in the Church of Rome, first, Linus [ordained] by Paul; and after him Clement, who was ordained by Peter.” …. The Ethiopic version, preserved by the Monophysite Church of Abyssinia, thus protected the traditional place of Paul in the Roman Church, which had been deemphasized since the beginning of the third century. [This document] reveals some knowledge of a relationship of both Peter and Paul to the Roman Church, but is not specific as to the character of such relationship. The two apostles are not mentioned specifically as either founders or bishops, but simply as apostles. … While the document is late and reflects use of the [apocryphal] Acts of Peter, the dual, undefined leadership of both Peter and Paul in Rome seems to be an echo of a second century tradition such as is found in Clement of Rome and Ignatius.
There are a number of things to understand from this:
1. O’Connor was aware of the “Catholic Hermeneutic” (““If the Party could thrust its hand into the past and say this or that even, it never happened–that, surely, was more terrifying that mere torture and death” — Orwell, “1984”)
2. The Ethiopian church was not aware of “Petrine primacy,” though it seems to have been aware of the “Catholic Hermeneutic.”
3. Note that Paul ordains Linus — he is typically second in any “Petrine succession” lists, and then later Peter ordains Clement. This directly supports Peter Lampe’s contention that the earliest church in Rome was not “ruled by Peter,” but rather, (a) there was a network of “house churches” in Rome, each of which likely had an elder or elders, who became the “Presbyterial-style government” that he wrote about, and (b) it also supports his contention that Irenaeus’s list from “Against Heresies” was a fictive construction, assembled from the names of known presbyters in that city (but that it was only later “assembled from memory,” not that there was any kind of intentional “historical succession”.
4. Those who find some kind of “petrine succession” in Clement and Ignatius aren’t reading those documents the way some in the earlier church read them.