You reject the evidence of Irenaeus and Cyprian then you say there is no evidence Timothy.
These Church Fathers were pre Constantine in the time of pagan Roman persecution, and and they looked at the Apostolic See as having primacy over all the Church.
The Chair of Peter is Gods guarantee of Teaching Authority and Truth, to the whole Church.
In protestantism people have no guarantee of the Infallible Truth, it's doctrines are divided an conflicted on the differing opinions of men.
Each denomination and non- denomination has a human founder and interpreter, and they are divided as a result.
The division is just a manifestation of the crisis of authority.
Just as the Church Fathers looked to the Chair of Peter on matters of doctrine, so a Catholic does today. Nothing has changed and it is entirely consistent.
"…I think it my duty to consult the chair of Peter, and to turn to a church whose faith has been praised by Paul…The fruitful soil of Rome, when it receives the pure seed of the Lord, bears fruit an hundredfold…My words are spoken to the successor of the fisherman, to the disciple of the cross. As I follow no leader save Christ, so I communicate with none but your blessedness, that is with the chair of Peter. For this, I know, is the Rock on which the church is built! This is the house where alone the paschal lamb can be rightly eaten. This is the ark of Noah, and he who is not found in it shall perish when the flood prevails.” Jerome, To Pope Damasus, Epistle 15:1-2 (A.D. 375).
The flood is the sea of error and opinion that exists outside The Chair of Peter, which is the Ark of Infallible guidance.
There were many who sneered at Noah's Ark, right up to moment the doors of the Ark were closed.
Genisis 6:21 "As for you, take for yourself some of all food which is edible, and gather it to yourself; and it shall be for food for you and for them."
" Simon, son of Jonas, lovest thou me? He saith unto him, Yea, Lord; thou knowest that I love thee. He said to him, Feed my sheep."
Just as God commanded Noah to feed His animals so Peter is also told to feed His Flock.