If you are baptized into Christ you become a member of His church which He established "on this rock" which has continued since His time. That church is called catholic or universal. Martin Luther began the great schism or protestant revolution and many small "c" christian churches resulted from that time. Like it or not as you say, there is still only one Catholic Church and it can trace itself back to Christ and his Church which he established on that rock, Peter.
The problem with your claim, is that I can read the words of Peter and see if what is call the Catholic Church of today is the same ‘church’ Peter founded. By the way Christ began the ‘protestant’ revolution when He cleansed the Temple. And He will yet again be doing some cleansing regardless of a church’s denomination. Christ warned what would be by those that sit in the seat of Moses, the law giver.. Reads just like the state of religion worldwide. Matthew 23 whole chapter. We are told to ‘test’ the fruit, and the present poppy is as far from Christ as any flesh being can become.
Maybe it can trace itself back through claimed unbroken succession to Peter, but that doesn’t mean those claims are true. It just means that someone managed to *find* an alleged unbroken line and voila`, succession.
Not good enough proof.