Who ever said that people that do not believe in flat earth do not believe in other conspiracy theories? There are plenty of people who think things like the JFK assassination, moon landings, Boston marathon bombings, etc have something much more to them than we are being told and still believe the Earth is Round. I personally see some decent evidence with the moon landings being faked and there was an actual motive to that. I am however not even 50% convinced of that conspiracy theory though, but there is a lot of doubt. There are things that governments do try to conceal, they are actually not that great at doing it though. Humans have a serious inability to keep secrets. The Flat Earth conspiracy motives, ability to "fake" the vast abundance of evidence of a round Earth, and keeping it a perfect secret for such long time just doesn't add up.
Flight times, weather patterns, GPS, Satellite TV/Internet, Sunset/sunrise, Actual area and shape of oceans and land masses, Southern circumpolar stars, Antarctic research stations, Seeing the ISS with a telescope, Solar/Lunar eclipses and their predictions, etc. Almost all of these things can be checked out by anyone without much money or time invested in them. Its just so hard to get my head around an entire global conspiracy for hundreds of years involving so many people to not have any random pilot, astronomer, flight attendant, Janitor at a research station, Meteorologist, Military person, etc. to have shown any evidence to the contrary that is actually verifiable.