I agree that Clement's letter should not be in the canon (the criteria for canonicity are very strict), and I agree that it demostrates the apostolicity of the Papacy. I don't see any circular logic though.
The circular logic of the Catholic Church is based on the assertion: "The Catholic Church is always right." It is often covered in several layers of abstraction, but it is always there none the less.
1 Clement starts the whole process and that's the real corner stone of the Papacy.
Catholics will claim it is the keys, but that's further down the circular logic track. First claim primacy, second claim the right to interpret the Bible, then claim primacy through interpretation.
Which came first, 1 Clement or the keys?
It reminds me of the Supreme Court. Some judges look to the Constitution for truth and principle. Other judges look to the Constitution to prove their own belief.
The keys given to Peter are an example of looking to the Bible to prove an existing belief.