For the case of UFOs, wouldn't we have to have scientific reports explaining all unexplained observations of aerial phenomena that have occured in history, before we could reasonably claim that the probability is very low?

No. There no reason to expect every observation to be explained. Tracking down causes of things is hard work and it's not reasonable that in every case someone did the hard work.

Using the word "scientific" in this context also illustrates a misunderstanding of what it means. You can't run scientific trials to find the meaning of a single event that happened in the past.