The ORAL Torah was later codified and expounded upon in the Talmud.
That’s a difficult contention, since clearly Jesus relied on that oral tradition contending with Sadducees over life after death. Ref. Luke 20:27-38.
Paul who is an apostle also numbered himself among pharasees phil 3, acts 22:2-5.
I would think rather that Jesus had his own interpretation of scripture which tended to coincide and agree with the prophets and their tradition. But which included critiques of their misapprehensions of God’s will.
Perhaps your claim about oral tradition entails something else apart from the prophetic books and the position the pharisees took.