Be it as it may, the letter of Clement does contain that claim implicitly, by referring to the rod of Aaron. It correct actions taken by local authority, condemns it, and insists on correction.
Even your best case scenario, which I don’t accept, would have the Bishop of Rome claiming authority. You do not have other Bishops from other regions accepting it.
We don't know at which point in time that was isnerted into the text or whether it was part of the original. But we do know that the First Letter of Clement appeals to reason, arguing that someone who is without blemish should not be deposed. But, the Epistle of Clement is not a pristine document either, as so many present it to be. For one, Clement repeatedly refers to the OT as scripture, but never to any references to the NT (mostly Pauline Epistles).
Here are more interesting facts about I Clement.
...translating presbyteroi as seniores (elders) rather than episcopi...
Pauline prayer that believers should submit themselves in all humility to the civil authority, has been reversed to state precisely the opposite: a prayer that all princes and rulers may now subject themselves to the Church, the Gelasian Pope Gelasius I doctrine that was being revived and put into effect during the eleventh-century Gregorian reform Gregorian Reform"
So, whatever we read in them, keep in mind that no matter what it says, it is also a "a modification of the text to suit the later spirit of the Roman church," something so often encountered with all documents of yore. The details of it shows that beyond any doubt (reversing the Pauline teaching on authority, etc.). Note also that it refers to bishops as "elders."
What I am saying is that although the authenticity of the original author is not so much disputed, the letter cannot be taken as Gospel, knowing that changes have been made in it.
Prudence therefore suggests proceeding with a grain of salt and not reading too much into it.