If a conspiracy exists, it would need to encompass people and organisations who are in a position to discover the truth, not just the ones who already know about it. This becomes problematic for a conspiracy model as, with such a fundamental truth like "the earth is flat" the number of people who could conceivably work it out becomes very large indeed. Especially since, according to the FAQ, anyone with a telescope and a big stretch of water can prove it.
Simply put, sure, the President doesn't 'need' to know the truth, but being the President and all, it probably wouldn't be too hard for him or her to find out, given that NASA is an agency of the US government. So there would need to be a pretty good contingency plan within the conspiracy to deal with that. Multiply that by every person who could work it out if they wanted and you get an awful lot of contingencies.
And it gets extra complicated when you consider that space race was an offshoot of the cold war, which means the Americans and the Russians were both faking aerospace data whilst covertly trading bullets in the Middle East and South America. And that's not even adding China into the mix, let alone Japan, India, the UK, Israel and Iran.
It's a soft spot that doesn't get much attention on this site, but the fact is that NASA is not the be all and end all of space exploration, and in truth, played catch-up to the Soviets for the majority of the cold war.