1 Clement nowhere demonstrates a single bishop - his name is not even in the epistle. Rather, it was from the elders, not from the bishop of Rome. The letter indicates there was no bishop of Rome at the time. It was later generations that ascribed it to him, singularly. If you read it you will realize it does not contain the sort of language popes used later on. It is exhortation, not commandment.
Now to this part of your post,
Re: dating of Clement
catholicnewsagency.com/column/from-peter-to-clement-to-benedict-1487/
If we look at Bp Irenaeus of Smyrna, today called Izmir (that’s in Today Turkey,)
Bk 3,
Chapter 3 , vs 2-3, ~180 a.d.
*
2. Since, however, it would be very tedious, in such a volume as this, to reckon up the successions of all the Churches, we do put to confusion all those who, in whatever manner, whether by an evil self-pleasing, by vainglory, or by blindness and perverse opinion, assemble in unauthorized meetings; [we do this, I say,] by indicating that tradition derived from the
apostles, of the very great, the very ancient, and universally known Church founded and organized at Rome by the two most glorious apostle , Peter and Paul; as also [by pointing out] the faith preached to men, which comes down to our time by means of the successions of the bishops. For it is a matter of necessity that every Church should agree with this Church, on account of its preeminent authority, that is, the faithful everywhere, inasmuch as the
who exist everywhere.
http://www.newadvent.org/cathen/01626c.htm
3. the apostles then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate. Of this Linus, Paul makes mention in the Epistles to Timothy. To him succeeded Anacletus; and after him, in the third place from the apostles, Clement was allotted the bishopric. This man, as he had seen the blessed apostles, and had been conversant with them, might be said to have the preaching of the apostles still echoing [in his ears], and their traditions before his eyes. Nor was he alone [in this], for there were many still remaining who had received instructions from the apostles. In the time of this Clement, no small dissension having occurred among the brethren at Corinth, the Church in Rome dispatched a most powerful letter to the Corinthians, exhorting them to peace, renewing their faith, and declaring the tradition which it had lately received from the apostles, …*
you can open the link
Chapter 3 ] and read the complete context
For some added reference:
Irenaeus was a disciple of Bp Polycarp, who was also a disciple of John the apostle
Polycarp [SAINT]