For me, I rather like science's approach. After all, it's the same science that gives all this technology that we are using as we speak. It's the same approach that has allowed us to put mankind on a celestial body other than our own, has landed amazingly complex robotic craft onto other planets and asteroids/comets, and has even put spacecraft beyond our solar system. I'd say that suggests that we may not know everything, by a long shot, but we've got a pretty good grasp on the important stuff.
Is science fallible? - yes.
Are there sometimes radical new developments? - yes. But they are getting much less common nowadays.
The thing is, science always uses - and progresses from - the
known. The unknown (eg from this thread - anti-gravity, 'inertia dampeners' and the like) is not accepted until it is tested and verified. What's more, if you allow all ideas, then everything is fair game - elves, faeries, pink unicorns, infinite multiple universes where all things are not only possible but inevitable, the 'matrix' theory that renders everything invalid anyway.. and so on..
Plus, there are some major issues with some of these ideas, eg anti-gravity and inertia dampeners are prime examples.. IF (and that's an
enormous IF) these things have been mastered, then by definition, the owners are able to generate free energy. Now you really need to think about that - if that's true, then why haven't they launched huge radio transmissions outwards? Why bother coming here unless to examine us primitives and perhaps give us a helping hand - we could not possibly be a threat nor could they need anything from us.. And why clumsily and accidentally reveal themselves in ways that are generally indistinguishable from distant aircraft, RC's etc (or swamp gas and balloons

)..?
I'm afraid the logical side of me can't accept the logic of aliens being here ...
and hiding.
I stopped doing that when I was a kid..
BTW, I'd be happy to look in detail at whatever case that shows
compelling evidence of 'outlandish maneuvers'. Let's get specific, shall we?