The reason I don't use secondary sources is because I rarely say anything that is controversial from the standpoint of mainline historians. (And if I did, primary sources would be the proper place to justify my unorthodox historical position.) That includes your complaint concerning papal primacy.
Primary sources are eyewitness or contemporary accounts. They are the main source of history. Secondary sources are the works of historians who have read or seen (in the case of archaeology) the primary sources. Tertiary sources would be writings of history based on secondary sources.
You claim that all historians acknowledge that Victor (bishop of Rome from A.D. 189-199) was a pope to whom the whole church submitted. Do you really think that Protestant secondary sources, in general, acknowledge Victor or any other second century Roman bishop as a pope? Protestant historians generally agree that Pope Gregory was the first pope, around A.D. 600.
Roman Catholic historians, obviously, almost always hold that the early bishops were full popes.
Some Roman Catholic secondary sources apparently agree with me, however, even though they wouldn't be able to use the same words I use. Dr. Brendon McGuire is a Catholic historian who lectures for the Institute of Catholic Culture. His lecture on the medieval papacy can be downloaded free here:
That lecture talks about the primacy of the bishop of Rome, and it discusses how no Roman bishop really had the power that Rome wanted to claim for the pope until Gregory. He explains how Gregory obtained papal primacy, which has nothing at all to do with Scripture or apostolic succession.
After writing this letter and posting it, I went back to listen to Dr. McGuire's lecture. The question is at the 49-minute mark of part one. His answer to the question of when the eastern churches acknowledged the pope's authority was not "never," but that the eastern churches have always seen papal primacy "differently" than Rome sees it.
I have adjusted my paragraph (next to this sidebar) accordingly.
Later, in the Q&A, someone asks him when the eastern churches acknowledged the pope's authority, and he honestly tells them that the eastern churches have never understood the pope's authority the same way Rome does.
Oddly enough, just today I ran across a Roman Catholic secondary source, a book on ecclesiology by Father Richard P. McBrien, the Crowley-O'Brien Professor of Theology at Notre Dame. He has also serves as the president of the Catholic Theological Society of America. He is also a syndicated columnist in the Catholic Press. I'm not sure how much more mainstream Catholic a person could be without being the pope himself!
His book, The Church, says the following on p. 98—note that this is immediately after he quotes Ignatius as saying Rome holds the chief place "in the territories of the district of Rome":
Remarkably, however, [Ignatius' letter to Rome] is the only one of Ignatius' classic letters to the seven churches of the ancient Mediterranean world that makes no mention at all of a local bishop. This lends credence to the supposition of historians and other theologians that the monoepiscopal structure of church governance (i.e. a diocese with one bishop as its pastoral head) did not even come to Rome until the middle of the second century, probably with the pontificate of Pius I (ca. 14-ca.155). [parentheses McBrien's]
That's one argument I made. There was no individual bishop in Rome until around the mid-2nd century. this Catholic historian acknowledges that's entirely possible. (He's already acknowledged that there's no set church structure in the New Testament churches.)
But he goes on. There's no break between the last quote and this one:
It would have been extraordinary, however, if Rome had not been singled out for a special role and position of authority in the early Church. Not only was it the place traditionally regarded as the site of the martyrdoms and burials of both Peter and Paul; it was also the center of the Roman Empire. Gradually Rome did emerge as an ecclesiastical court of last resort, the local church to which other local churches and their bishops would appeal when disputes and conflicts could not be settled between or among themselves. For example, in the controversies with Gnosticism ... defenders of orthodoxy appealed to the faith of episcopal sees founded by the Apostles, and especially to the faith of the Roman church because of its close association with Peter and Paul. ...
The correlation between Peter and the Bishop of Rome, however, did not become fully explicit until the pontificate of Leo I ... in the mid-fifth century (440-461). ...
With the beginnings of the East-West Schism in 1054, the shape of the papacy changed even more significantly. Before the split the Bishop of Rome had been viewed primarily as the patriarch of Rome alongside the patriarchs of Constantinople, Antioch, Alexandria, and Jerusalem. After the split, however, the Roman patriarchal office and the papal office merged. The patriarchal office was completely absorbed by the power of the papacy. ... The Bishop of Rome came to regard himself, and be regarded as, the universal primate of the universal church. (italics are McBrien's, bold is mine)
So what did I write about the church of Rome in the second century? I wrote:
[Irenaeus] really meant that every church should agree with Rome. That is the context of his whole argument. He is arguing that the apostles established churches, and then they appointed elders in those churches. To those elders they left a "tradition," a body of truth that the early church called the rule of faith. It was the elders job to preserve that rule: "We refer [the Gnostic heretics] to that tradition which originates from the apostles, which is preserved by means of the succession of presbyters in the churches." (Against Heresies III:2:2)
So why Rome? Because it would be tedious to reckon up the succession of all the churches, and besides, Rome was founded by the two greatest apostles. (ref)
Again, the wording may be a little different, and the emphasis may be different, but what I've said is not inaccurate, nor even speculative. It's sitting right there in history so plainly and clearly that even Roman Catholic scholars have to acknowledge it.
As an interesting note, this person wrote back and complained that I only cited one secondary source to him (although I gave him two). Yet his email, claiming that ALL historians agreed with him cited only one source and that without a quote.