metis
aged ecumenical anthropologist
The vast majority of Jews and Gentiles may have been illiterate country bumpkins but the authors of the Gospels were certainly not. These men were highly educated Jews.
Actually most Christian theologians that I have read would disagree with you, with maybe the exception of Paul and possibly James, If you have a chance, read Martin Marty's "The First Christians" or Dr. Hansons "Tradition In the Early Church" Either way, it's highly unlikely they had their own Torah to refer to, so their references would most likely have been added later.
Each Jewish group undoubtedly claimed as such, much like we've seen happen over and over again with Christian denominations. It's the "You don't have it right, but I do, so follow what I believe" approach.Their goal was not to reinvent Judaism but to renew it. They saw the end of the Temple period, not the end of Judaism.