Generalized what quote? I see none.
I’m sorry, but it’s beginning to seem to me as if you have no idea what you’re talking about and are losing track because of an apparent inability to distinguish either cite from anecdote, nor an historical cite pertaining to the church from scripture itself.
There were several churches early on, they’re referenced by name in the Epistles. The Apostle Paul even wrote to Rome and instructed the church there, rebuked it even. Rather odd, don’t you think, if Peter were Pope, for Paul to have done such a thing?
Because the epistles/letters were written before Peter went to Rome. He established his presence later in Rome.