Precisely so. Also, Pope Clement made the scriptural foundation of his primacy very clear as he compared the twelve Apostles to the twelve tribes of Israel where Aaron had primacy:
[...] when rivalry arose concerning the priesthood, and the tribes were contending among themselves as to which of them should be adorned with that glorious title, [Moses] commanded the twelve princes of the tribes to bring him their rods, each one being inscribed with the name of the tribe. And he took them and bound them [together], and sealed them with the rings of the princes of the tribes, and laid them up in the tabernacle of witness on the table of God. And having shut the doors of the tabernacle, he sealed the keys, as he had done the rods, and said to them, Men and brethren, the tribe whose rod shall blossom has God chosen to fulfil the office of the priesthood, and to minister to Him. And when the morning was come, he assembled all Israel, six hundred thousand men, and showed the seals to the princes of the tribes, and opened the tabernacle of witness, and brought forth the rods. And the rod of Aaron was found not only to have blossomed, but to bear fruit upon it. What think you, beloved? Did not Moses know beforehand that this would happen? Undoubtedly he knew; but he acted thus, that there might be no sedition in Israel, and that the name of the true and only God might be glorified;
So Clement laid the foundation for the Papacy. This is exactly my point. Here is the start of the circular logic. After you unwind all of the layers, you end up with one man writing a letter that was rejected from the canon.
Two interesting things about 1 Clement. First, it claims inspiration (1 Clem 63:2). Second it speaks of the phoenix in Arabia as a real bird (1 Clem 25).
1 Clement is not canon for good reason.
I think this will come in handy too, thanks.