Wait, the claim is that the past 60 years or so of space exploration/industry is all fake.
That is a claim made and defended by many. However it is not required for the world to be a different shape than we are taught. What makes the faq perspective on "the conspiracy" novel, is that it is suggested the faking/hiding the shape of the earth is not the reason for any of that. The cold war is chock full of propaganda, of which "the space age" is only a part.
Yet the services offered actually work (the 4 different GPS system, broadcast TV, telecommunications (some still remains), landsat, weather sats, all the imagery (of the earth as well as the other planets and their moons) and so on). So for the real thing to be fake and for some other method to be provided for providing all that CORRECT (i.e. consistent with everything we know) data, it would take 10s, if not 100s of thousands of folks across many countries. How is that not a conspiracy?
However, the services are not related to space travel (they are intentionally conflated with it). They are demonstrably real. How they actually work/function may be misunderstood or otherwise wrong. It is extremely unlikely, but even something that appears to us as "orbit" may work in an entirely different manner than is currently thought. We have no trouble being wrong without help from "conspiracy"; we've been doing it since the dawn of time.
The presumed curvature of the world seen in photos is a good example. We expected to see a curve, so when we saw one from high altitude photography we all concluded "that's it, just what we expected!". Even today there are many who still erroneously believe the horizon curves at some altitude, because of this simple misunderstanding (of lenses/optics/photography) and misinterpretation of data - taught to generations of students. Many things we see and take photos of are not truly as they appear, and our interpretations of them are even more consistently incorrect.
An impossible one I'd say (both due to fact that so many folks would not have any hope of keeping any secret so well and due to the need for some alternative to providing all the data).
Keeping secrets is all too easy, especially if you include death in your recipe/procedure. In any case, most of the employees in the space industry have no secret to share - they are simply employees who do a job. If there were any sort of conspiracy it would be well beyond their pay grade. Very few people would need to keep any secret, and this is one of the major purposes of compartmentalization. Another, is to prevent any one employee/asset/soldier from seeing and understanding the larger strategy/motive/purpose employed.
The data you mention, by and large, is not fabricated - it is collected by some real means and then composites are created from it. Few, if anyone, suggests that the data is being made up wholesale.
The disk earth simply can not explain the universe we observe.
This is wrong, but I understand why you feel that way. The shape of the earth doesn't play a significant role in science or our understanding of the universe. Also, we by and large don't observe the universe - we observe the earth and only experience life thereon. The shape of the world is flat either because the world is flat or because it is effectively flat on the scale we live and practice science. A disc earth is not an insurmountable impediment to explaining anything.