No, fundamental Christian theology is based on grace and salvation, and the redemptive power of the crucifixion and resurrection of Christ.
Luke and Mathew - the only gospels which briefly mention a birth narrative - say nothing directly about Christian practice and theology.
The gold standard for Christian practice, theology, and salvation are the epistles of Paul. Paul said nothing of consequence about Jesus' birth, and apparently considered it so inconsequential it doesn't factor in at all Paul's instructions and guidance on Christian practice, theology, and salvation.
Paul knew the apostles Peter and John, and he knew Jesus' brother James, so if the birth was so important they would have told him.
There is decent circumstantial evidence that Mark's gospel is based on Peter's teachings. Peter was Jesus' closest apostle, and the birth narrative is never mentioned in Mark. If the birth was so bloody important, you'd think Peter would have told Mark.
Now, the birth story of the Buddha is even more fantastical than the birth story in the Gospel of Mathew. But it doesn't figure prominently in the practice, beliefs, and metaphysics of Buddhism.
Atheists have never used the Buddha birth story to diminish Buddhism.