Conspirers are everywhere. Moon landings could not have been faked, in my humble opinion. If you lived through the cold war era, you should understand the chest bumping between the East and West and the constant monitoring of each others activities. Just saying..
A lot of the accusations were about the levels of technology available at the time. Sure, the rockets had as much memory as a desk calculator, but if you're using a fairly simple system (compared to modern rockets and shuttles) then it was enough.
Don't get me wrong, going to the moon was hard, but it wasn't
that hard. Propulsion, aerodynamics, controls, oxygen, living quarters and a pod with reverse thrust. The difficulty was doing it for the first time, and they had enough casualties and failures to get to where they are now.