tptacek is right that invocations of Galileo of that sort are usually a trope but a great thing about the trope (and about Galileo!) is that it almost always falls apart and argues against the point the trope-invoker is making. In your case - Galileo did not 'find the rings of Saturn'. He made some rather anomalous observations of Saturn which he could not explain. He published a 'claim hash' (in the parlance of our times) of his observation.
Galileo was one of the earliest practitioners of what we now call scientific method and even with that disadvantage, he was way better at simply making claims than the UFOlogists we're talking about here.
Let me simplify the idea so that we don't get caught up in historical debates: we won't move beyond anecdotal evidence of UAPs unless our academic institutions start looking for them systematically and/or we reach out to our elected representatives to declassify/release whatever data the DoD has already collected.
