Who was responsible for maintaining the Christian oral tradition?
The apostles, and after them the successors appointed by them.
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.)