No, the keys are an ancient symbol of authority and have nothing to do with a Berean like quest for verification of beliefs in Scripture. This is particularly evident since at at the time of the "keying" there was only OT Scripture.
The idea that the keys gave Peter primacy was first mentioned by Optatus, who lived in the fourth century. This is an example of looking back into the Bible to get what you want out of it.
Getting back to 1 Clement...
We know that the letter contains factual errors. We also know that this letter lays the foundation for the papal succession. Should we believe the letter contains doctrinal truth?
Here comes circular papal logic to the rescue. According to Papal Infallibility, all Popes are protected by the Holy Spirit from doctrinal errors. So 1 Clement contains doctrinal truths even though it contains factual errors. Therefore the papacy is established.
However, 1 Clement also claims inspiration. This is a problem. Works inspired by the Holy Spirit are protected from all errors. However, 1 Clement contains factual errors so it can not be inspired. So if 1 Clement claims to be inspired and it is not then it contains doctrinal errors. Therefore the doctrinal foundation of the papacy is in question.
1 Clement is the base on which the Papal House of Cards is built. The Church is built on the unshifting rock. The Catholic Church is built on 1 Clement.