The theoretical possibility of FTL drive has never been a major factor is discounting crazy stories of alien encounters. The total lack of physical evidence and unreliability (or "craziness") of supposed witnesses is what invalidates most of it.
First off, I wasn't talking about recent alien encounter stories, which are mostly discounted by the fact that are constantly watching the skies and we would have noticed anyone landing. In fact, except for the Wow! signal, any specific
claim of alien contact is probably bullshit and could be right only by chance – these days anything is attributed to aliens, so if something was actually done by aliens, no doubt someone would have attributed it to them without actually having a clue.
That being said, a huge part of the reason alien stories are that
crazy is because we are rightfully assuming that there aren't any aliens anywhere close enough to visit us, both in distance and price of the journey. If that is not the case, the odds will be significantly changed, and while many of those stories will still remain crazy, they will be less so, and many would no longer be.
This is particularly true for the more general ideas such as paleocontact or the possibility of alien spacecraft remains in the Solar System, etc. And if someone tells you they've dug out a fossilized warp engine from their backyard, you'd at least
give them the benefit of the doubt to click on the link of the picture. Right now, I would not even keep on reading what they have to say.