Like so many statements that are misleading, this can be understood in a way that makes it accurate.
1. It is true that, if Christianity has a single "hinge," it is the Resurrection of the Jesus Christ.
2. It is true that Christianity does not depend on the "stories around the birth of Jesus," especially any "stories" that go beyond the texts of the Gospels.
3. And finally (the really slippery part), "the truth ... around the birth of Jesus" is not as much of a critical point as the celebration of Christmas makes it seem. It is Jesus's conception, of a virgin mother by the Holy Spirit, that makes Him true God and true man. His birth is just a move from one place to another.
However, taken all together, the specific events of Jesus's conception, birth, and infancy are both very important as the detailed fulfillment of prophecy and as symbols of what is to come.
What evidence do you have that your assertions in points 1 and 2 are true ?
It’s not the birth. It’s the gospels.