In the interest of paying attention to what those on the ‘other side’ think about the narratives of Jesus’ birth as found in Matthew and Luke, I note the following from Jonathan Pearce.  Some of the objections are trivial.  Others merit more careful consideration.

The only general point that needs to be made here is that no-one is required to be able to answer all possible objections before an event can be considered historical.

Anyway, here’s what Jonathan Pearce writes:

‘In order for the Christian who believes that both accounts are factually true to uphold that faithful decree, the following steps must take place. The believer must:’

On the virgin birth:

• ‘Special plead that the virgin birth motif is actually true for Christianity but is false for all other religions and myths that claim similarly.
• Deny that “virgin” is a mistranslation.’
• ‘Give a plausible explanation of from whence the male genome of Jesus came from and how this allowed him to be “fully man”.’

On the genealogies of Matthew and Luke

• ‘Be able to render the two genealogies fully coherent without the explanation being contrived or ad hoc.’
• ‘Believe that the genealogies are bona fide and not just tools to try to prove Jesus’ Davidic and Messianic prophecy-fulfilling heritage.’
• ‘Be able to explain the inconsistency of the two accounts in contradicting each other as to where Jesus’ family lived before the birth (without the explanation being contrived or ad hoc).’

On the census:

• ‘Somehow be able to contrive an explanation whereby Herod and Quirinius could be alive concurrently, despite all the evidence contrary to this point.’
• ‘Believe that a client kingdom under Herod could and would order a census under Roman diktat. This would be the only time in history this would have happened.’
• ‘Find it plausible that people would return, and find precedent for other occurrences of people returning, to their ancestral homes for a census (at an arbitrary number of generations before: 41).’
• ‘Give a probable explanation as to how a Galilean man was needed at a census in another judicial area.’
• ‘Give a plausible reason as to why Mary was required at the census (by the censors or by Joseph).’

On the journey from Nazareth to Bethlehem:

• ‘Give a plausible explanation as to why Mary would make that 80-mile journey on donkey or on foot whilst heavily pregnant, and why Joseph would be happy to let her do that.’
• ‘Believe that Joseph could afford to take anywhere from a month to two years off work.’
• ‘Believe that, despite archaeological evidence, Nazareth existed as a proper settlement at the time of Jesus’ birth.’
• ‘Believe that the prophecies referred to Nazareth and not something else.’

On the magi:

• ‘Believe that the magi were not simply a theological tool derived from the Book of Daniel.’
• ‘Believe that Herod (and his scribes and priests) was not acting entirely out of character and implausibly in not knowing the prophecies predicting Jesus, and not accompanying the magi three hours down the road.’
• ‘Believe that the magi weren’t also merely a mechanism to supply Herod with an opportunity to get involved in the story and thus fulfil even more prophecies.’
• ‘Believe that the magi were also not a reinterpretation of the Balaam narrative from the Old Testament, despite there being clear evidence to the contrary.’
• ‘Believe that a star could lead some magi from the East to Jerusalem and then to Bethlehem where it rested over an individual house and not be noted by anyone else in the world.’

On the shepherds:

• ‘Believe that the shepherds were not merely midrashic and theological tools used by Luke.’

On the different witnesses (magi and shepherds) in Matthew and Luke:

• ‘Believe that there is (and provide it) a reasonable explanation as to why each Gospel provides different first witnesses (shepherds and magi) without any mention of the other witnesses.’

On the Massacre of the Innocents:

• ‘Believe that, despite an absence of evidence and the realisation that it is clearly a remodelling of an Old Testament narrative, the Massacre of the Innocents actually happened.’
• ‘Believe that Herod would care enough about his rule long after his death to chase after a baby and murder many other innocent babies, a notion that runs contrary to evidence.’
• ‘Believe that God would allow other innocent babies to die as a result of the birth of Jesus.’

On the Flight to and from Egypt:

• ‘Believe that the Flight to and from Egypt was not just a remodelling of an Old Testament narrative in order to give Jesus theological gravitas.’

On the apparent discrepancy about where Jesus and family went after his birth:

• ‘Give a plausible explanation as to why the two accounts contradict each other so obviously as to where Jesus and family went after his birth.’

On the disappearance of the shepherds and magi from subsequent history:

• ‘Explain the disappearance of the shepherds and magi, who had seen the most incredible sights of their lives, and why they are never heard from again despite being the perfect spokespeople for this newfound religion.’

On the unbelief of Jesus’ own family, in the light of the nativity accounts:

• ‘Provide a plausible explanation as to why Jesus’ own family did not think he was the Messiah, given the events of the nativity accounts.’

Pearce, Jonathan M.S.. The Nativity: A Critical Examination (pp. 128-130). Onus Books. Kindle Edition.

Note: I find that I have given some thought and study to most of the issues raised in the above list.  Kindly see the linked articles.  There are a few objections (such as the absence of the magi and shepherds from subsequent history) to which the only sensible response is: ‘absence of evidence is not evidence of absence’).