I think it would be an enormous change. Just look at how much more harmless things through history has changed everything. Some monk nailed his opinion on the church door and everyone lost their shit. It is probably the best parallel to an UFO disclosure. People got radicalized just by realizing that they've been lied to (which I assume many already understood, but was afraid to say).
In this case there is a lot that is different too, since we already have a lot of narratives in place (maybe by design). Even if the revelation is just that "aliens are here and flying around in saucers", people will still jump to the conclusion that USG and other countries has kept it secret. Which leads to the idea that the whole history since the 40s has been fake, and so on. Add to that all sorts of much darker ideas about children being intentionally abused to develop psi-powers or getting trafficked, as Barber claims, or the recent talk about it being demons and that USG has a pact/deal with them, and maybe that some groups, classes, or ethnicities are actually alien hybrids, and so on and so on.
Luther's harmless opinions resulted in the downfall of the Christian unity in the west and religious wars and so on, and that was just about the question about authority, this is more like as if he had proven that the Pope was the devil, or something else that destroyed the whole foundation for their faith and worldview.
Of course, it might be that we are too damaged and hypnotized by social media to realize what is important or not, so that society has a different resistance to shocks like that compared to back then.