This story, most famously told by Jehovah Witnesses and Fundamentalist Protestants, came out of their necessity to support their lie that there was an apostasy in the early Church. It is their way to explain how their reform and late arrival is justifiable.
I woke up on the morning of October 31, 2003 as the same agnostic/deist that I had been for most of my adult life, but by the end of that day I would actually end up praying to the God who I was unsure even existed.
Indeed, throughout the Gospels we clearly see Jesus affirming the proper application of the Oral Torah; that is, where it does not contradict or attempt to supplant the Written Torah. This is especially clear in many of Jesus' dialogues with the Pharisees in response to His works done on the Sabbath.