It's a little more absurd even than that, Joe. By the time "Luke" was written....whenever the hell that was... the author did not know that Bethlehem (in Judaea) had been in a different country than "Nazareth" (which most likely did not exist at all) but was in Galilee in any case. As you say, Archelaus was deposed in 6 AD and Quirinius as the new governor of Syria was directed to incorporate Judaea as a prefecture into his command. However, Galilee remained under the rule of Herod Antipas until 39 AD when Herod Agrippa I was made king of both regions by his boyhood friend, Gaius (Caligula.) Thus, for the entire period of time covered by "Luke" the notion of someone going from Nazareth to Bethlehem would be like an American going to Canada for a census. Laughable but explainable in the sense that the division of the territory actually lasted for only a short period of time in the early first century. For whatever reason the Romans were always eager to find a member of the Herodian clan to run the place for them. When Herod Agrippa I died in 44 it ushered in a period of direct Roman rule under a series of procurators while the Romans waited for Herod Agrippa II to become old enough to be invested as king.
Thus, perhaps the author of "Luke" can be forgiven for not knowing that bit of historical minutiae concerning the political realities of Judaea/Galilee in 6 AD but he still fucked up the story.
Thus, perhaps the author of "Luke" can be forgiven for not knowing that bit of historical minutiae concerning the political realities of Judaea/Galilee in 6 AD but he still fucked up the story.