The biggest problem I see for a flat Earth, at this point, is the size of the necessary conspiracy. Just the number of people involved in the space program is staggering. The people that have been to space, the people that put them their, the people that work with the data gleaned from there, the industrial applications, the satellite shots, the whole GPS thing, the engineering, the various private programs that utilize GPS, telecommunications, the international efforts to compete in conquering Earth's orbits...
If the Earth is flat, and all these people are either complicit in the hoax, or deluded AND (Somehow!) prevented from realizing the hoax, or asking difficult questions, or publishing problematic findings...
And that's just the space side. The people navigating the waters around the South Pole would have to be in on it, on how the Earth's circumference at Southern Latitudes is so very much larger than at Northern latitudes; pilots; cartographers; weathermen...
Well, it can't really be called a conspiracy, can it? Because at that point, it would just be the state of the industry. And the state of the science. It would hardly qualify as a conspiracy if more people are in on it than are fooled by it.