It's my position that NASA truly and honestly believes that the earth is a globe, which is why they chose to depict it as such in their Apollo images. But since neither space travel or earth orbit is possible, and since the highest NASA can go is to the edge of the atmosphere where they are looking down at the distant curved edges of the sun's circular area of light, they are in no position to tell us what form the earth takes.
The "conspiracy" revolves around faking space travel for political and military purposes. After the Manhattan Project in the 1940's the US needed some kind of unstoppable vehicle to facilitate the transport of its weapons of mass destruction. Thus NASA was created in the 50's for the sole and only purpose of convincing the world that the US could drop an ICBM on foreign cities at the drop of a hat.
You're fooling yourself if you believe that NASA was created for some kind of humanitarian or scientific purpose.