I am not expecting or trying to persuade anyone about Matthew's gospel, but I hope to make it more historically plausible for all readers, believer and non-believer.
First, Christianity and Islam are peculiar, in that they are both historical religions. Both depend upon a human who appears in historical times and who does confirmable events. Both religions rely upon accounts and revelation for their polity. Other religions, like the pagan, might have had historical roots (Euhemerism maintains that the gods were heroes whose graves were worshiped and thus were called on in times of distress), and Buddhism has the historical Siddhartha, but the precise historicity of Siddhartha is not necessary for the practice of the religion. Christianity and Islam cannot function without Jesus and Mohammed.
There are three features of Matthew's nativity that are unique to his account -- features that today we have folded out into legend. The first is the star of Bethlehem, then visit of the magi, and the slaughter of the holy innocents is the other. These are of a piece and Matthew's.
People should believe as free will dictates. However, those who attack Christianity on historical grounds make me scratch my head. From Pliny the Younger on, Christianity was under attack by the Romans, and it was earlier under attack by the mainstream Jewish world, and these pressures were contemporaneous with the Gospel writers. If there were substantial doubts about the existence of Jesus, wouldn't one of those groups -- both of whom had the records that we now try to reconstruct -- have launched the attack? If the followers were making empirical claims that could be disproven, wouldn't contemporaries have jumped on it, when the evidence was fresh?
That said, Matthew's account sticks out. Luke's does, too. Both have unique material. Luke has the shepherds in the fields. It's Matthew, though, who makes the more historical claim.
What I would like to offer, below the in hoc signo lexit, is a way for a non-believer, a historian, and a believer to all get along with Matthew's account. In other words, I want to offer a way for us to all just please get along.
A bit after it came out, I read Adrienne Mayor's The Poison King, which is a biography of Mithridates (ok, I waited until after it won the National Book award). I had been a classicist for a while, and I'm still interested, so I was interested in this figure who was always so alien and "weird" in Latin accounts (Livy linked above, and the new Carthage Must Be Destroyed will explain to you why he posed as Heracles so often). I recommend the book highly, if my recommendation is necessary for anyone.
What stuck out to me, though, was the background it offered on just who the heck these "Parthians" and "Medes" were of that era. In what follows, you are better off reading Mayor or a scholarly source than a web source, but I will try to link for validation. Mithridates was born in 134 BC, and he was born under a comet. Tigranes III was also born under the comet, and both used the comet as their official symbols. You see, in Zoroastrianism, a comet is the fire of the heavens, and it marks the birth of a Great King.
The Mysterious East
At the time of the three wars Mithridates waged against Rome, the remainder of the Hellenistic world was shattered. Rome had been picking off the Great Powers. Although Rome doesn't seem to us now to have had a grand plan, they might as well have had, because the Macedonian, Antiochan, and Ptolemaic empires were clobbered or weakened one by one. The Seleucid was all that was left, and that grand Eastern grouping of nations was the remnant of the Persian empire.
Mithridates did something no one thought possible: he made alliances with the Scythians. These people (today's Georgians) were fierce horsemen. He also gathered tribes of "amazons" (also Caucuses groups) and brought in people from the most impossible place of all, Bactria (Afghanistan). He got Tigranes to be a subordinate in a combined war, as well.
How did he do this? He took on a conventional eastern title: "King of kings." We can think of that as analogous to "commander in chief." In other words, he was a king commanding other kings for a grand military adventure to repel Rome from their lands.
For the Zoroastrian peoples and the many, many kingdoms of the East, Mithridates was a savior. He was a unifying force who stood up to (and did not loose to) Rome. He expelled the Roman tax farmers.
So, a "Star, a Star?"
There are people who insist that the Star of Bethlehem had to be a super massive object in outer space some light years away that had nuclear fusion which appeared for a short time and then disappeared, I'm sure. Some will, I am sure, demand that it be a divine will o' the wisp, too. However, a comet makes a lot more sense.
If a comet appeared, the Medes and Parthians would have noticed. As Zoroastrians, they watched the skies, and they would have sought out a comet. If that comet appeared over a "house" in the sky, they would have known instantly that it meant that a new savior was born.
...A new Mithridates, to unite the eastern kingdoms in overthrow of Roman oppression.
They would have hustled out, too. They would have been prepared, as well, for the birth to be here or there, as the royal house of David would be the royal house of David, and Roman taxation forcing people to be in a shed instead of a palace.... Well, things are tough nowadays for royalty.
Matthew's Gospel is more rabbinical than Luke's, and even casual readers notice all the emphasis on prophetic fulfillment in it, and that makes it a bit harder to causally read its nativity. However, I recommend looking again, because the Parthian wise men who come are not given any number (three? fifty? just "some") and the gifts are not comprehensively listed. Matthew gives the gifts as symbols of royal anointing and priesthood.
IF star THEN innocents?
The major problem with the comet is why we don't have records of it aside from Matthew. We have Chinese records, else we'd have none at all.
This merely begs the question, though, of why we don't have records of any other comets, either, or novae or lunar eclipses. The people who kept astronomical records tended to live in the places where records got destroyed. Think about your glib and glum "history of astronomy" pellet handed out in elementary and high school: The Greeks figured out the Ptolemaic spheres in spite of data to the contrary, and then Copernicus.
There is another reason: religion.
Specifically, Roman religion saw comets as omens of doom. If there was a comet, it meant that someone great was going to die or that the military would suffer a huge loss or that there would be a natural calamity. The Romans got seriously freaked out by comets. (A tertiary reference to Ian Ridpath (1985) here gives, from the Roman Marcus Manilius, a long list of things that comets indicated, including blighted crops, plague, wars, insurrection, and family feuds.) Thus, if the Romans saw a comet in 5 B.C., they would not be too eager to write about it, celebrate it, or go looking for a baby under it.
However, let us consider Herod the Great for a moment. He was not a well loved king.
John the Baptist would illustrate just how unhappy religious and nationalist reformers were with him and his policies, but even aside from the religious forces Herod was a client king. He was not from any of the royal families, did not have the levitic blessing, and did not care. He was king by the grace of Rome, not God. He had his title and power due to Roman soldiers, and the people did not forget it.
So, if you were Herod, and some leaders or ambassadors from the various kings of the many, many nations to your East came along and said, "Hey, a new Mithridates has been born right here -- one who will get the title of king of kings and lead us in overthrowing Roman rule," how would you react?
Herod was vulnerable. The Romans would put him in power, but a confederation of eastern kingdoms willing to mass and take on Rome herself? Well, that calls for a military solution, doesn't it?
Historical context and religion
If what I have written makes anyone feel that I am demystifying or secularizing the nativity account, it was not my purpose. Our mistakes are sometimes our greatest deeds, and sometimes astrologers thinking they're finding a new Mithridates end up finding a greater thing. After all, they would not be the only ones to look for Jesus as a political leader and find instead a spiritual one.
Instead, I merely want to open Matthew's story to some degree. If the comet appeared, then the rest is historically logical. It was always religiously and spiritually true, and its truth has never been altered by political context, I hope.