One line of history is that the Apostles which ministered with Jesus leave a written record of And there your history stops, until 1500 or so years later.
When something stops, it doesn't continue, that's discontinuity.
The true church has always (through the centuries) ... rested upon the foundation of the teachings of the Apostles (whether in their lifetime or from their writings).
I don't believe that the claim is that there was no time when the histories of the Catholic and the Protestant and the Orthodox churches were joined.
I think you made the case for a major disjoint occurring right after the first Apostles.