If the government admitted that they had faked all space programmes it would make them look bad I guess, Flat earth thinks all space exploration is faked, the reason NASA gave us round images of earth is because that what we expected to see. Plus governments get so used to lying it becomes part of their nature, they probably couldn't honestly tell you what they had for breakfast.
The problem with this theory is that the flaws in the Flat Earth hypothesis would have been VERY obvious to the captains and navigators of square-rigged ships going back to at least the 1600's. They'd have known for sure if the real world didn't fit the RE maps they were given.
Yet there is not a single example of a book on celestial/compass navigation from any time in those last 400 years that mentions the problem.
If the Earth is indeed flat - then those navigators, captains and cartographers would have HAD to know about it...for 100% sure.
So it could only be that they had some kind of duplicate set of navigation charts and manuals that told them how to get around in the flat earth - and yet not one single copy of any of those things has ever been found.
This monumental, 400 year conspiracy would DWARF anything that NASA would need to hide.
You might argue that governments of the time might want to hide this fact (WHY?!) - but that's not enough. We know that civilizations like China and the Arabian countries of the time were also sailing long distances using celestial navigation. Even before we contacted them, they were using round earth navigation techniques.
But yet more than that - what about the Pirates? They too sailed long distances using compass and stars to navigate, they had no great love of governments - yet not one of their books (of which there are many) ever refers to having to know the secret art of flat-earth navigation.
When you look back into history - the conspiracy would have to be impossibly wide/deep.
Claiming that NASA are responsible would be merely touching the tip of the iceberg of this conspiracy.