"Christ did not just tell them how it was, He told them to search the scriptures for themselves."
John 5:39 - But when Jesus said "search the Scriptures," He was rebuking the Jews who did not believe that He was the Messiah. Jesus tells them to search the Scriptures to verify the Messianic prophecies and His oral teaching, and does not say "search the Scriptures alone." Moreover, since the New Testament was not yet written, the passage is not relevant to your claim of sola Scriptura.
"And despite this common claim by Catholics that private interpretation leads to division, the Catholic church itself even with its authorities and all is actually quite fragmented and full of little sects and divisions within itself. "
The Catholic Church considers itself a single Church ("one Body") composed of a multitude of local or particular Churches, in each of which the one Catholic Church is embodied. However, the Catholic Church attaches great importance to the particular Churches within it.
A particular Church, in Catholic theology and Canon law, is any of the individual constituent ecclesial communities "in full communion with Rome" that are part of the Catholic Church as a whole. Canon 368 of the Code of Canon Law.
They are: Armenian Catholic Church, Belarusian Catholic Church, Bulgarian Catholic Church, Byzantine Catholic Church, Chaldean Catholic Church, Coptic Catholic Church, Croatian Greek Catholic Church, Ethiopian Catholic Church,
Georgian Catholic Church, Greek Catholic Church, Hungarian Catholic Church, Italo-Albanian Catholic Church, Latin Church, Febronianism, Gallicanism, Maronite Catholic Church, Melkite Catholic Church,
Romanian Catholic Church, Russian Catholic Church, Ruthenian Catholic Church,Slovak Catholic Church, Syrian Catholic Church, Syro-Malabar Catholic Church, Syro-Malankara Catholic Church, and the Ukrainian Greek Catholic Church