"Actually, there is no record of Linus claiming to be anything. What is recorded is that in 180, Irenaeus wrote that the apostles committed to Linus the office of the episcopate."
Linus was made a Bishop by the Apostles, which was the leader of a local congregation.
Somewhere down the road the Catholic church decided the head of the church would be the "Bishop of Rome".
List of Popes
Peter was an Apostle. Linus was not. If you look at the list provided by New Advent, John was probably still on the earth even through Pope Clement I.
Linus, Anacletus, and Clement (I) literally could
NOT be head of the true Church of Christ, one because they never had the authority and two, a higher authority was
proven to be on the earth at the same time.
Linus was made a Bishop by the Apostles, which was the leader of a local congregation.
Somewhere down the road the Catholic church decided the head of the church would be the “Bishop of Rome”.
I agree that Linus was only the leader of the episcopate of Rome at the time of Peter’s death. I also agree that it was somewhere down the road that the Bishop of Rome was given primacy. My point was in response to your statement that Linus claimed to be the head of the Church. I am not aware of there being anything in the historical record that shows that Linus claimed anything; that we really don’t know anything about him.