Beating a dead horse, but remember the Manhattan Project?
Every measure was taken - compartmented knowledge, fake names, false paper trails, actual fake towns and industrial sites were built with acted communities and workers. Censored mail, extensive surveillance, coded communiques, and more that I cannot remember off the top of my head.
It was probably the most extensive attempt by any government to keep a project secret.
... And it still leaked. Multiple times. The secret couldn't be kept. The sheer scale of it and the human element, the fact that other countries learned enough to be able to find even that minimal paper trail.
All that before the age of instant worldwide communication.
Today, even harder to keep that kind of thing secret despite (and because) of the technology leaps. And the sheer thousands of people that would have to be involved. And global suppression of flat earth is magnitudes larger in scope than the Manhattan Project.