But of course alien civilizations—or (non-human?) co-terrestrial civilizations—are such an extraordinary hypothesis that I do not see why they should be treated as more plausible than (as you note) something like shared delusions.

Let me expand upon that a bit. My (tongue in cheek) explanation is this:

Humans have an undiscovered facility for shared perceptual delusions and (seemingly deliberate) confabulation. That is to say, individuals with shared cultural contexts may respond to the same environmental triggers (weird reflections, social stressors, I don't know!) by hallucinating the same things. Moreover, upon experiencing such a hallucination, some subjects will confabulate corroboratory evidence, perhaps even (unconsciously) going so far as to tamper with radar recordings or (unconsciously) colluding with other "witnesses"—and individuals who engage in this confabulation have no memory or recollection of it later.

This is a totally wild explanation, and I don't actually believe it. But is it as plausible as alien civilizations? I sorta think so!