Does the background to First Clement support a papacy?

First Clement opens with a reference to how the Corinthians had requested help from the Romans, how the document was being written to address "the points respecting which you consulted us" (1). It's sometimes suggested that papal authority is implied. Why did the Corinthians consult the Romans? Since there's a good chance that the apostle John was still alive, why didn't they consult him instead?

People frequently consult a lesser source rather than a greater one for a variety of reasons. Sometimes, the lesser source is consulted because he's lesser. Why burden somebody as important as John when a lesser source, like the Roman church, could assist you instead? That sort of thing is a common practice. Or the lesser source might be more familiar. If the Romans and the Corinthians had been in frequent contact before the problem in question arose in Corinth, the Corinthians might have turned to the Romans for help due to familiarity, much as you'd often turn to a friend for help rather than turn to a stranger, even in contexts in which the stranger has more to offer in some significant ways. There's also the possibility that John was unavailable due to his not being alive at the time, his being in too weak a condition to warrant consulting him, or his being unavailable for some other reason or too occupied with something else at the moment, such as traveling.

Then there's the fact that we have many other circumstances like the ones behind First Clement that don't lead us to think a papacy is involved. We even have that in another context involving the Corinthians around the same time. Why did they consult Paul to ask him questions rather than consulting Peter (1 Corinthians 7:1)? Polycarp wrote of how the Philippians had "invited" his counsel (Letter To The Philippians 3). Why didn't they contact the Pope instead? Maybe Paul and Polycarp were Popes. Of course, it would be ridiculous to reach that conclusion. It's also ridiculous to reach the conclusion of a Roman papacy based on the similar situation with First Clement.

In reality, First Clement is evidence against the papacy and Roman Catholicism more broadly. The document makes many and explicit appeals to various authorities (the Father, the Son, the Holy Spirit, scripture, the apostles, elders and deacons, etc.), but never makes any appeal to a papacy. It's written in the name of the church of Rome rather than the bishop of Rome. And it's most naturally interpreted as contradicting Roman Catholicism on a range of issues, such as justification.

Other early extrabiblical sources are likewise problematic for Catholicism: Josephus, Ignatius of Antioch, Aristides, etc. We're often told that scripture is unclear, how we need to look to extrabiblical sources to interpret scripture for us, and so forth. When we get to the extrabiblical sources, they also provide evidence against Catholicism rather than providing evidence for it.