The Bomb was a game changer, everyone working on the Manhattan Project knew it, and they _still_ had spies successfully infiltrating the project and exfiltrating above-top-secret data. Some supposedly alien material wouldn't change human nature. Something that big would leak.

It did leak. Everybody "knows" the government has been experimenting on Aliens and alien technology since at least Roswell. There are even articles in the New York Times about warehouses storing alien artifacts!

That is a good point. If you were in possession of world-changing, society-breaking, panic-inducing information, how else would you prepare society for eventually hearing the full truth?

You would do it as a laughable joke over a long period of time.

That is why they stopped trying to outright hide projects like that, and instead mix the truth in tsunami of disinformation, sensationalize, crack pot theories, conspiracy, and the like so even if legitimate "Top Secret" info was leaked it could be safely dismissed as conspiracy, lies, or the newest term "fake news"