Ok. Because if you read it, Clement calls James that “Bishop of Bishops” in the opening. This is a bit odd if Clement understood himself as the universal shepherd head of the church universal.
So I am not sure how to understand what is happening in Rome’s bishopric from the beginning. If Linus is the first to succeed Peter, and then Cletus is 2nd to suceed Peter, why is it said that Peter ordained Clement? Did Peter ordain all three of them while he was living? If so, are they successors or Co-Laborers? This is getting a bit disorganized, and at best, a bit skewed compared to modern day definitions of how the papacy has been since the apostles…
I think you should go back to the quote you provided…"But before dying, taking by the hand his disciple S. Clement, S. Peter appointed him his successor, an office which S. Clement would not accept nor exercise till after the death of Linus and of Cletus, who had been coadjutors of S. Peter in the administration of the Roman bishopric.
Coadjutors are assistants or auxiliary bishops. It looks like Linus and Cletus were bishops ahead of Clement, before Peter ordained him as a bishop.
Upon the death of Peter, Linus would ascend as head bishop…then the auxiliaries would be Cletus and Clement…then Cletus, his auxiliary would be Clement.
I think you also may be presuming them as power hungry men…when in fact, Clement’s humility is shown in the quote…by deferring to Linus and Clement.
I think this quote from Clement’s epistle to Corinth should help you understand…where he writes about this procedure…
earlychristianwritings.com/text/1clement-hoole.html
42:3 They, therefore, having received the promises, having been fully persuaded by the resurrection of our Lord Jesus Christ, and having been confirmed by the word of God, with the full persuasion of the Holy Spirit, went forth preaching the good tidings that the kingdom of God was at hand.
42:4 Preaching, therefore, through the countries and cities, they appointed their firstfruits to be bishops and deacons over such as should believe, after they had proved them in the Spirit.
42:5 And this they did in no new way, for in truth it had in long past time been written concerning bishops and deacons; for the scripture, in a certain place, saith in this wise: I will establish their bishops in righteousness, and their deacons in faith
44:1 Our Apostles, too, by the instruction of our Lord Jesus Christ, knew that strife would arise concerning the dignity of a bishop;
44:2 and on this account, having received perfect foreknowledge, they appointed the above-mentioned as bishops and deacons: and then gave a rule of succession, in order that, when they had fallen asleep, other men, who had been approved, might succeed to their ministry.