If Jesus was born outside Bethlehem, would the ancient sources be so unsupportive of that conclusion?

Critics of the Bethlehem birthplace often act as if they're confident that Jesus wasn't born there or that he was born in Nazareth instead. I've addressed their arguments many times, such as in the posts collected here. The article here argues for the likelihood that the early Christians and their opponents had access to reliable information on Jesus' birthplace and the likelihood that they obtained that information. For a brief overview of the evidence for the Bethlehem birthplace, do a Ctrl F search on "shows" here and go to the last hyphenated section here for information on ancient non-Christian sources. What I want to focus on in this post is something I wrote in a Facebook thread a few years ago. This is about whether evidence was lost or suppressed for a false date Jesus and the early Christians had set for the second coming. The same principles can be applied to the notion that Jesus was born outside Bethlehem, but that the evidence for that birthplace was lost or suppressed:

Furthermore, you would have to argue that a Christian willingness to suppress the material in question was as widespread as your hypothesis would need it to be. We don't begin with a default assumption that so many sources were of that character. Ambrose, Origen's patron, asked him to respond to Celsus on every point (Against Celsus, Preface, 3), something Origen thought unnecessary, but said he would do anyway (Preface, 3; 2:20; 2:46; 4:6; 8:76). He doesn't respond to everything in Celsus' treatise. There are places where he refers to passing by something Celsus wrote because it's too repetitious, for example. But given how much Ambrose wanted Origen to respond to and how much he said he would interact with, even where he'd prefer not to, it's unlikely that he would have ignored as significant an objection as the one we're discussing in this thread. Where Origen does interact with Celsus' material, he sometimes acknowledges that Celsus' objections are significant, doesn't have much of a response to what Celsus argued, etc. (e.g., 2:63)…

The evidence we have pertaining to ancient Christianity isn't limited to material the ancient Christians expected to be made public. Rather, much of what we have consists of private letters, documents and archeological artifacts that were rediscovered after having been lost for centuries, etc. Christians wouldn't have been trying to avoid discussing matters as significant as what we're addressing in this thread in all of those contexts….

And, as I said before, you need to argue for such a low view of the character of the early Christians rather than just asserting it. When modern Christians are troubled by something in scripture or some other relevant source, some of them try to avoid discussing it, but many do discuss it. We'd expect that kind of variety during other eras of history as well, not the sort of widespread silence we see about the alleged false prophecy of Jesus and the earliest Christians.

Remember, the question in this context, as in others, is which explanation of the evidence is best, not just whether an explanation is possible. If there are two or more explanations that are possible, then you can't decide among the explanations or justify agnosticism about them merely by saying that one or more of them is possible. You have to go beyond that.