To me what is unreliable is the problem with reading comprehension of Luke 2:2 by ignoring the word " first ".
'First' would indicate there was another registration at another time.
Also, a person can also be governor for a 'second' time.
There is No evidence that Luke's account was ever challenged by the earlier historians including Celsus.
No, Judaea was a client kingdom, and Roman census would only occur if Judaea was annexed as a Roman province, after Augustus had Archelaus removed from his throne and exiled, which was 10 years after Herod’s death, as recorded in Josephus’ Antiquities of the Jews.
Rome don’t do census on client kingdoms. Herod and his son were client kings to Augustus, they paid tributes, not taxes to Rome.
Plus, Quirinius was legatus of Galatea from 12 to 1 BCE, trying to quell insurrection in Cilicia and Cappadocia. He wasn’t legatus of Syria until 6 CE, after Archelaus’ banishment. When Judaea became province, Quirinius became its governor (proconsular legatus), had control of the legions, and had more power than a prefectus of Judaea, so Quirinius was appointed to initiate the Roman census.
At the time he was governor of Galatea, Syria was contemporarily governed by the following governors when Herod the Great was still alive (till 4 BCE):
Marcus Titius (13 - 9 BCE)
Gaius Sentius Saturninus (9 - 6 BCE)
Publius Quinctilius Varus (6 - 4 BCE)
Quirinius was only legatus of Syria from 6 to 12 CE, 10 years after Herod’s death in 4 BCE. And Archelaus was king of Judaea from 4 BCE to 6 CE.
When Herod died, Augustus divided the lands between Herod’s sons, with Galilee belonging to Herod Antipas, who along with Archelaus in Judaea, he became client king, as tetrarch. Antipas ruled for 43 years, until he himself was banished in 39 CE.
Beside that, in Matthew 1 & 2, it conflict with Luke 1 & 2.
In Matthew, Joseph was already a resident in Bethlehem, have a home in the town when Jesus was born. No where in Matthew 1 & 2 Nazareth and Galilee were mentioned until 2:22-23, when they left Egypt:
“Luke 2:22-23 said:
22 But when he heard that Archelaus was ruling Judea in place of his father Herod, he was afraid to go there. And after being warned in a dream, he went away to the district of Galilee. 23 There he made his home in a town called Nazareth, so that what had been spoken through the prophets might be fulfilled, “He will be called a Nazarene.”
The gospel of Matthew not only never mention anything about Nazareth and Galilee, until this passage, BUT MORE IMPORTANTLY, this gospel never mention anything about any census or any governor of Syria.
In Luke 1 & 2, it showed that Joseph and Mary as residents of Nazareth, and since Joseph was living in Galilee, this conflict with Matthew 1 & 2.
You would only need to register in Roman census, where you were residents of, not where Joseph’s ancestors and ancestry home were.
According to Luke 1, both Joseph and Mary were already residents of Nazareth, and in Galilee, it was Rome’s client kingdom, where Antipas was client king from 4 BCE to 39 CE.
Whoever wrote the gospel of Luke, was terribly unreliable. Josephus was contemporary to the authors of both gospels, but he came from noble house, an aristocrat, who joined the rebellion in 65 to 70 CE, but was captured and became hostage of Vespasian, and befriended Vespasian’s son Titus, the future emperor.
As an aristocrat and friend of Titus, Josephus had both Jewish and Roman sources, hence he would be more reliable than the 2 gospel authors. And Josephus was definitely more reliable as he mentioned both Saturninus and Varus as governors of Syria, that’s in agreement with Roman records. There are no records that two census were carried out in Judaea while Augustus was alive.
Quirinius was never mentioned by Josephus till Archelaus was banished from Judaea in 6 CE.