That isn't true - the Church teaches "by the Teachings of Christ and the Apostles alone," so to speak. We include Sacred Tradition with Sacred Scripture, both sides of the same coin - the Oral teachings and the Written teachings. The Book of Mormon is an attempt to create new Scripture, while Tradition is unbroken since the time of the Apostles.
This idea of independent-authoritative Tradition (undefined) is an OPEN DOOR to innovation and additions to the teachings of the Apostles--contained in Holy Scripture. Many if not most Roman Catholic distinctives are based on Traditions that have no record before the 13th Century.
I must admit I fail to understand why Roman Catholics cannot see their's is an organization based on revisionist history without a solid foundation--as amorphous Tradition can be almost whatever the current Pope and Curia wants it to be.
Once you make a foundation God's Word AND the traditions of men--EXACTLY what the Pharisees of Jesus day did--one has an open door for the corruptions of men to undermine the solid foundation of the very words of God.
The whole concept of "private interpretation" too is a bogeyman. If an "every man for himself" teaching was a part of Protestantism there would be no Protestant Creeds--and yet almost immediately, both Luther and later Calvin developed creeds--that is detailed catechisms for their Church of what the Bible teaches. Calvinist and Lutheran creeds too are remarkable for all of what they have in common....probably over 95% overlap. Why is that? They trusted God's Word as the only sure foundation.
The reason we have thousands of denominations is that unlike the Middle Ages--with truly a Roman Catholic, government enforced religious monopoly, we have freedom of religion...
The only way one would not have breakaway denominations is to return to the days of the Inquisition--and I trust most Roman Catholics would not want that.