This documentary claims the authors of the Gospels were contemporaries of Jesus. That Matthew and John were actually amongst Jesus disciples. That the Gospels are based on eyewitness testimony.
Well, I'll look at the documentary, as it's only 10 mins long, but first of all let me just say that it's not very credible to suggest the Gospel writers were all eyewitnesses, as they contradict each other and sometimes themselves on important details. For example:
In those days a decree went out from Emperor Augustus that all the world should be registered. This was the first registration and was taken while Quirinius was governor of Syria. All went to their own towns to be registered. Joseph also went from the town of Nazareth in Galilee to Judea, to the city of David called Bethlehem, because he was descended from the house and family of David. He went to be registered with Mary, to whom he was engaged and who was expecting a child. (Luke 2:1–7).
"Jesus was born in Bethlehem of Judaea in the days of Herod the king" (Matthew 2:1)
Luke also talks of Jesus’ birth in the context of: “in the days of Herod, the king of Judaea” (Luke 1:5).
And yet, Quirinius’ Census (which did take place) was held in AD 6-7, when Herod had already been dead 10 years. If Matthew and Luke were both there, they are mistaken about something fairly fundamental - Herod can't have been King at the time of the Census. So that puts Herod's role in question, along with the Slaughter of the Innocents.
Come to that, the Slaughter of the Innocents was only mentioned by Matthew. None of the other gospel writers thought it worth mentioning. Nor, indeed, does any contemporary source outside of the Bible. The first century Jewish historian, Josephus, for example, is silent on the matter, despite an interest in Herod’s brutality.
Compare also Matthew 2:15, 19 and Matthew 21-23 with Luke 2:22 and 39 on what the Holy Family did after the nativity.
Matthew’s narrative has them going from Bethlehem to Egypt and then Nazareth. Luke, however, has them going from Bethlehem to Jerusalem and then Nazareth.
Matthew:
where he stayed until the death of Herod. And so was fulfilled what the Lord had said through the prophet: “Out of Egypt I called my son”.
After Herod died, an angel of the Lord appeared in a dream to Joseph in Egypt
So he got up, took the child and his mother and went to the land of Israel. But when he heard that Archelaus was reigning in Judea in place of his father Herod, he was afraid to go there. Having been warned in a dream, he withdrew to the district of Galilee, and he went and lived in a town called Nazareth. So was fulfilled what was said through the prophets, that he would be called a Nazarene.
Luke:
When the time came for the purification rites required by the Law of Moses, Joseph and Mary took him to Jerusalem to present him to the Lord
When Joseph and Mary had done everything required by the Law of the Lord, they returned to Galilee to their own town of Nazareth.