There is no evidence of such extensive "backfilling." That article you posted mentions...let's see...some question about dates, family names, birthplaces, in one case the order of office. But as far as the overall list and its fundamental integrity, that you can find in any secular encyclopedia because it rests on solid historical evidence. Let's run through a tiny bit of that evidence for a moment.
Eusebius, writing soon after A.D. 300, has the following in Chapter 3 of his History of the Church (I'm cutting and pasting here from various sections):
After the martyrdom of Paul and of Peter, Linus was the first to obtain the episcopate of the church at Rome. Paul mentions him, when writing to Timothy from Rome, in the salutation at the end of the epistle....After Vespasian [*the Roman emperor] had reigned ten years Titus, his son, succeeded him. In the second year of his reign, Linus, who had been bishop of the church of Rome for twelve years, delivered his office to Anencletus.....In the twelfth year of the same reign Clement succeeded Anencletus after the latter had been bishop of the church of Rome for twelve years.And he continues on from there. We actually still have a letter written by Clement, so arguing he was made up would be a little strange. And if you think Eusebius was too late of a source, I can take you back to St. Irenaeus writing around A.D. 170-180 or so. Irenaeus as a boy knew St. Polycarp, who was a disciple of the Apostle John--so his testimony is only a few degrees away from the Apostles themselves. Here's from Book 3 of Irenaeus' Against Heresies:
The blessed 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....To this Clement there succeeded Evaristus. Alexander followed Evaristus; then, sixth from the apostles, Sixtus was appointed; after him, Telephorus, who was gloriously martyred; then Hyginus; after him, Pius; then after him, Anicetus. Sorer having succeeded Anicetus, Eleutherius does now, in the twelfth place from the apostles, hold the inheritance of the episcopate. In this order, and by this succession, the ecclesiastical tradition from the apostles, and the preaching of the truth, have come down to us.
There is no need to continue this discussion because I won't change your mind and you definitely won't change mine. :-)
As for the Clement Letter To The Corinthians: It is a fine pastoral message, certainly from one who commands respect. I can find no indication that it approaches the level of a Papal letter to a subordinate Church. Can you?