The hypothesis that the Earth is round pre-dates NASA by some way. Though its true that the hypothesis that the world is flat pre-dates it by more. I would argue that there is enough evidence to argue this point without using satelite photos, space travel etc. The flat earth hypothesis deviates most obviously from the spherical earth hypothesis in the geometry of the Earth's surface. There are I believe many commercial flights per day around he southern hemisphere, in planes with well characterized performance capabilities, even if these were being lied about we can trivially tell that their maximum speed is less than that of sound. Yet the time taken to travel between say Buenos Aeries and Sydney is not compatiable with the FE geometry. Also someone looking out of the window would probably have questioned the route by now as in the FE world the shortest route from Australia to South America is via the North Pole.
Though I would argue the strongest evidence comes from the fact the the world is not as opaque is people often think. As I pointed out in a previous post, the Earth is almost transparent to Neutrino's. By detecting neutrinos we can see straight through the Earth. Indeed artificial beams of neutrinos are routinely fired into the Earth an detected where they re-emerge. All things considered I would conclude that the RE hypothesis is more consistent with data that I have observed.