Interesting theory in J.P.Holding’s book “Christmas is Pagan and Other Myths” which I hadn’t run across before, that helps reconcile differences in the Birth narratives in Matthew and Luke. It does make sense.
We know that the Gospel of Matthew was composed in Antioch, Syria. If members of the Magi (who could have been from Syria, not Persia according to some scholars), either the aged originals who journeyed to Bethlehem, their sons, or members of their priestly line with whom the oral tradition was passed down, were Matthew’s informants, they would have imparted knowledge that only they would be likely to know - about the events and conversations in Herod’s Court. They would not have had knowledge of the events on the night of Jesus’ birth (which would have occurred about two years before their visit), and if told about the shepherds, would not have considered their witness as important as members of a priestly aristocracy (rather like being told a garbage man or maid was a witness, in their eyes.) They would not know about the Roman census (as they were in a non-Roman country), or anything about John the Baptist, or the twin pregnancies of Mary and Elizabeth, or Zecheriah.
Matthew alluded only obliquely to Mary’s visit from the angel (she “was found to be with child through the Holy Spirit”), but then, Matthew’s audience would be less likely to credit the witness of a woman, Mary. Luke shows great regard for women and their thoughts (possibly due to his profession as a physician, and the greater experience he had in tending to women’s needs and suffering), was more likely use them as informants, and focuses more on Mary and her cousin.
Luke, who was writing a Gospel that was “Roman Friendly” for the gentiles, and to convince the Romans that the new Church was no threat to them, would have based his narrative on Jesus’ and John’s family sources including Mary and others. The Magi were from the Parthian Empire, one of Rome’s deadliest enemies, so he would be unlikely to mention them - or the flight to Egypt (Rome’s other biggest foe). (It’s been suggested that the Gospel of Luke was written as a court brief for Paul to use in trial before a Roman magistrate.) Luke would not have wanted to show Jesus as receiving gifts from the emissaries of a hated enemy, or taking refuge in the country of another enemy from a puppet king of the Roman Empire (Herod).
The Magi would have been, as members of another patriarchal culture, likely to have spoken directly to Joseph and obtained information from him rather than Mary (then as now in the mediterranean, one did not speak directly to young women if not a family member, in most circumstances, out of delicacy and respect.) In that culture, it would have been expected that the paterfamilias would have been the one to receive the honor of the first angelic vision, so perhaps Joseph did not want to share that information with the Magi.
Luke may or may not have known about Joseph’s vision, but as his focus was on women, he may not have included it (or Joseph had passed away by this point, and was not available to Luke as a source.) Luke does not record any major objections by Joseph, which is a little unusual unless there was something he knew about that convinced him that Mary’s honor was not in question.
Luke also seems to know something that Matthew did not - Nazareth was the actual hometown of Mary and Joseph. This would be a detail he would know from family informants, but the Magi might not have known, or they just presumed the family was originally from Bethlehem, or they were simply not told by Joseph (Joseph might not have been comfortable sharing that detail with the Magi - he may not have known if they were trustworthy or what they might tell Herod.)
The flight into Egypt, and the events afterwards, would be unlikely to be known by the Magi after they returned to Syria, so it is unikely that this was information Matthew received from them. Most probably this was information Jesus told directly to his disciples. Luke would also have wanted to avoid allegations that Jesus learned magic in Egypt, which was an accusation later used by Roman critics against the Christians.
As Luke did not include the flight into Egypt, he could not mention the killing of the Innocents, because it made no sense unless in the context of that event.
To make a very secular comparison, I own about 30 biographies of Houdini, and it’s interesting to compare how different biographers describe different period in his life - birth, childhood, the vaudeville years, etc., based on which source each biographer interviewed. There’s rarely a conflict, but different biographers may not include certain events because sources were unaware of them, or because they had different foci.