So let's go with the thesis that they were written in this order: Mark, Matthew, Luke then John. It's all a bit of a speculation about precise dates, generally, but relative to each other and the destruction of Jerusalem by the Romans this is about right. Read in that order, the four books give these accounts of how Jesus arrived.
Mark: Announced by cousin John out in the wilderness. Cue baptism and a ministry that begins with a call to repent.
Matthew: A genealogy, angelic visitations, a parade of visitors, a holiday in Egypt and then in the wilderness with John.
Luke: A genealogy, angels, prophecies by elder Israelites, prophecies by cousins, more visitors, and so on.
John: Forget the virgin birth, Jesus is the incarnation of the divine logos itself. His cousin said so.
Two of these books don't even have a Christmas story. Of the other two, one is more elaborate than the other. Clearly, the events hold different value for each of the authors. The Christmas story is not a unified whole, a single story about a baby in a manger.
Matthew and Luke did us a favour with the additional information, for sure, but most of the time I think I'd be happy with the introduction from Mark, and put John a close second. There's something appealing to me about the Jesus who bursts on to the scene and just starts preaching.