Not familiar with Lampe's work, but it seems from your description like he is taking facts that everyone already knows and drawing exactly the wrong conclusions from them.
Of course the Roman church was an assemblage of house churches! Where else could they worship? There were no basilicas, and the catacombs were not yet being used. They would have gathered in the homes of sympathetic families. Excavations under the Basilica of St. Clement in Rome have shown exactly such a house church, later converted to a Mithraeum and then back to a Church. It is thought that this house belonged to the family of Flavius Clemens until they fell out of favor with the Emperor in the 90s.
And we already know that there were multiple bishops simultaneously. I don't remember which ancient author stated this offhand, but Peter was said to have personally ordained Linus, Cletus, and Clement to the episcopacy while he lived. Clement writes to the church in Corinth and says "WE feel that WE have been tardy in turning OUR attention to the points respecting which you have consulted US". He's speaking for a group.
But if you think this militates against any kind of presidency, then take a little trip to the chancery office of any diocese in this country, where you will find a Bishop served by several auxiliary bishops. The Pope is surrounded by cardinals, most of whom are bishops. But not a single one of them is at all confused about who presides. And the Apostolic succession passes through all of them not only as individuals but as a community.
The authority of the Church of Rome concerns less its internal organization and more its external relations. Ignatius tells us it "presides in Love". Clement tells us "If, however, any shall disobey the words spoken by Him through us, let them know that they will involve themselves in transgression and serious danger."
It seems every time we bring Irenaeus up someone can be relied on to reply "well, he's not inspired...he is subject to error". No kidding! But you'd think that if he were wrong on this, others who came after him would say "wow, man, you really went off the deep end there." That's exactly what happened to Origen and Tertullian. Who told Irenaeus he went off the deep end? Who told him he made a mistake, or he was only talking about Gnostics or whatever? Were is the sed contra from any Church Father?
And this business about "both Peter and Paul" founding the Church arguing against the primacy of Peter. Paul says he is not worthy to be even called an Apostle, whereas Peter was the head of the Apostles. Peter walked with Christ whereas Paul got knocked off a horse much later--do you think so little of your ancestors that they would not appreciate that distinction, even as they revered both?
You should familiarize yourself with Lampe before dismissing him sight unseen. The significance of his work is precisely that he marshaled an array of evidence unparalleled by previous efforts. Papal supremacy and succession are historical truth claims. If the only evidence for such claims are bare assertion by those not contemporaneous to the events, then those claims cannot be taken seriously as proof. Irenaeus at best is providing late, biased, and demonstrably erroneous testimony, again, not through ill will, but through the limits of circumstance. All of which makes an extremely poor factual foundation for making a claim of global authority.
Peace,
SR