The RCs state that their church was, and thus they are the supreme authority on what it means. By the same logic, the Jews of Christ's time were to submit to those who sat in the seat of Moses, as Israel with its magisterium "were committed the oracles of God," "Who are Israelites; to whom pertaineth the adoption, and the glory, and the covenants, and the giving of the law, and the service of God, and the promises." (Romans 9:4)
Why then did they follow a man in the desert who lived off of insects, and an Itinerant preacher, both of whom were rejected by those who sat in the seat of Moses?
Scriptures and the virtue and power of God it speaks of. Which the church also was established by. (Mt. 22:23-45; Lk. 24:27,44; Jn. 5:36,39; Acts 2:14-35; 4:33; 5:12; 15:6-21;17:2,11; 18:28; 28:23; Rm. 15:19; 2Cor. 12:12, etc.)
It is very important to understand 'tradition' from an Hebrew aspect, because the verse you allude to is precisely the thing that cuts the papists off at the knees.
The New Testament books are essentially the tradition of a Rabbi named Yeshua. It is the work of His disciples to preserve their beloved Rabbi. That requires a further Hebrew definition: what exactly is a disciple?
A male child of 6 years would have fully accomplished the Hebrew alphabet and prayers, mostly by his mother... The father then begins to teach the boy - and by the age of 14 he has memorized the entire Torah. If he has not done this with accomplishment, he would generally begin apprenticeship in his father's business. But if he excelled, he was encouraged to continue to memorize the entire Tanakh - whereupon he would approach a Rabbi to prove himself worthy (study to shew thyself approved)... The Rabbi would question him with great precision, and being satisfied with the answers (a rare thing), the boy was allowed to sit at the feet of the Rabbi. If not, then back to apprenticeship in his father's business.
It was an extraordinary honor to be a disciple of a Rabbi, and the greater the Rabbi, the more prestige. That Yeshua found his disciples working for a living means they had already failed to live up to the standard. But like with any Rabbi, being asked was a great honor, and like anyone asked, they would literally drop everything to follow the Master... Even for an itinerant one.
But what does a disciple DO? He emulates his Master. Every word is recorded and preserved... memorized. Everything the Master does, every action, is emulated and preserved. A disciple would rather cut out his tongue than speak a word against the words of his Master. These things - the actions and the sayings of the Master - are preserved inerrant, and are to be followed perfectly, or one is not a disciple of the Master. THAT is the tradition that the disciples of Yeshua are talking about.
So we come to this:
Mat 23:2 Saying, The scribes and the Pharisees sit in Moses' seat:
Mat 23:3 All therefore whatsoever they bid you observe, that observe and do; but do not ye after their works: for they say, and do not.
The scribes and Pharisees claim the right to Moses' seat - They claim to be the disciples of Moses - But one is not to pay attention to their works (their actions and sayings, supposedly in the service of Moses), because they 'say and do' *not* like Moses. So the only way this can be read is that the authority remains in Moses, to listen to HIM, and one must only listen to His disciples if they are under that authority - saying and doing LIKE MOSES.
Likewise, those who claim to be disciples of Yeshua have the authority of Yeshua ONLY if they say and do like Yeshua... They must believe (hear and do) in their Master, or they are false disciples, and are to be ignored.
This is an incredibly bold concept, turning rabbinic precedence and tradition on it's head... It prevents the accumulation of traditions, and leaves the authority where it belongs - in the Master, not his servants. This is why Paul corrected those who said they were disciples of Peter, James, John, etc... He was following the example of His Master - they cannot be disciples of the servant anymore. they must be disciples of the Master.