The sources, or rather channels, of public revelation are Bible, Tradition, and the Magisterium. Private revelation adds no doctrine at all.
It would be circular reasoning to use private revelation to support the Magisterium, since the Magisterium approves private revelation.
Stronger argument would be besides the Bible quotes, just the existence of a single, small Canon of the NT.
A second argument is the near universal acceptance of at least the basic core beliefs of tradition, such as the Trinity.
Without a papacy there might have been 300 books in the NT, and still adding them now. There might have multiple Christianities as there were in ancient times rather than the single family of similsr groups we have now.