The fact of matter that Judaea wasn’t a Roman province when Herod was still alive, because Judaea was a client kingdom, not a province. Romans don’t do census on client kingdoms, as Judaea was at the time. Census were reserved for provinces.
Second. Quirinius couldn’t have been governor of Syria as Gaius Sentius Saturninus (9 - 7/6 BCE) & Publius Quinctilius Varus (7/6 - 4 BCE) were successively governors during Herod’s last years. Josephus had even mentioned Varus as governor when Herod died.
while Saturninus and Varus were governors, Quirinius was serving as governor (pro praetore legatus) of Galatia (10 - 1 BCE), trying to quell rebellion mountain tribes that were hiding on the mountains of Galatia and Cilicia.
Quirinius only became governor 10 years later, when Augustus turn Judaea into province, after banishing Herod Archelaus. Then, and only then, was census carried out in 6 CE.
Third, the story of census where Joseph must go to Bethlehem, his ancestral home, is a myth. people were only needed to register in census, in their current home, so if Joseph was living in Nazareth, then Joseph didn’t need to register, as Galilee was a client kingdom , not a province at the time.
You don’t understand Roman imperial governing of Roman provinces any more than whoever wrote the gospel of Luke.
unlike ancient Israel, Judah & Judaea, Rome have far more contemporary texts and records, and their records provide insight how the Romans run Rome and the Roman Empire, politically, legally, socially & military.