So the church at Rome is attested long before Paul writes and there is no leader there.
Ignatius, who knows and writes about Bishops in the east, writes to Rome without mentioning a Bishop. There is no question the city of Rome is important. It is the capital of the empire. This church “which presides in the place of the district of the Romans…”
Shepherd of Hermas: According to the Muratorian Canon, the oldest (ca. AD-180-200?) known list of the New Testament and early Christian writings, Hermas was the brother of Pius, bishop of Rome (ca 140-154). So he was writing earlier than Hegesippus, whose “list of bishops” is said to be the first one (c. 166), and earlier than Irenaeus (c.180). Hermas was, in fact, listed in the Muratorian Canon as a book to be read in the churches [i.e., it was liturgical].
As I slept, brothers and sisters, a revelation was given to me by a very handsome young man, who said to me, “Who do you think the elderly woman from whom you received the little book was? I said “The Sibyl.” “You are wrong,” he said. “She is not.” Then who is she?” I said. “The church,” he replied. I said to him, “why then is she elderly?” “Because,” he said, “she was created before all things; therefore she is elderly, and for her sake the world was formed.”Roger Collins, “Keepers of the Keys of Heaven: A History of the Papacy,” (New York: Basic Books, 2008), notes “The author of the Epistle of Clement may have been the man of this name later described as the person responsible for drafting communications sent behalf of Christians of Rome to other churches.” If this Clement did compose 1 Clement, then it certainly would be understandable why the Corinthian church would have thought they received a letter from Clement (even though the name of Clement does not appear within that letter. Rather, it is from “the church of God that sojourns in Rome”).
Afterwards I saw a vision in my house. The elderly woman came and asked me if I had already given the little book to the elders (presbuteroi, plural). I said that I had not given it. “You have done well,” she said, “for I have words to add. So when I finish all the words they will be made known to all the elect through you. Therefore you will write two little books, and you will send one to Clement and one to Grapte. Then Clement will send it to the cities abroad, because that is his job. But Grapte will instruct the widows and orphans. But you yourself will read it to this city [Rome], along with the elders (presbuteroi) who preside (proistamenoi – plural leadership) over the church." (Vis 2.4)
But Hermas could not be more clear. There is a plurality of presbyters who “preside over” the church at Rome. This is no fuzzy mention, as in Ignatius, of a church in “a place of honor”. This is a clear explanation for the “argument from silence” in Paul’s letter to the Romans, in the absence of a clear leader in both 1 Clement and Ignatius.
Hermas reiterates the structure of this leadership, and the fact that they are not leading, but rather that they fight among themselves. He calls them “children”.
Look therefore to the coming judgment. You, therefore, who have more than enough, seek out those who are hungry, until the tower is finished. For after the tower is finished, you may want to do good, but you will not have the chance. Beware, therefore, you who exult in your wealth, lest those in need groan, and their groaning rise up to the Lord, and you together with your good things be shut outside the door of the tower. Now, therefore, I say to you [tois – plural] who lead the church and occupy the seats of honor: do not be like the sorcerers. For the sorcerers carry their drugs in bottles, but you carry your drug and poison in your heart. You are calloused and do not want to cleanse your hearts and to mix your wisdom together in a clean heart, in order that you may have mercy from the great King. Watch out, therefore, children, lest these divisions of yours [among you elders] deprive you of your life. How is it that you desire to instruct God’s elect, while you yourselves have no instruction? Instruct one another, therefore, and have peace among yourselves, in order that I too may stand joyfully before the Father and give an account on behalf of all of you to your Lord.” (Vis 3.9)Hermas here is chastising the multiple leaders of the church at Rome. This is important to note because Hermas identifies himself as a slave (Vis. 1.1). It will not do to say that this is a group of priests who work for a bishop. The entire group "presides."
Yet here, in the leadership of the church of Rome, there are multiple elders who "preside"; they are acting like sorcerers. They exult in their wealth. They take the seats of honor. They want to teach, but they are guilty themselves of having no instruction.
This is very clear writing. Very clear reporting of what the church was like. For those of you who want to understand what the leadership structure of the church at Rome was like, it is hard to find a better primary source witness than Hermas. [Please pay no attention to the fact that there are many scholars whose work corroborates what Hermas says here. This is a post about primary sources.]