I believe it is just another case of the yanks using publicity to prove a point to the world. No doubt they did make it to the moon, but I believe that film was shot in a studio.
As we are all aware, they have proven over the years that twisting publicity is one of their strong points. No conspiracies there I am afraid. Anyone who has been to the Pearl Harbour will have noticed lots of signs around the place reminding the world that WW2 started in 1941 after the Pearl Harbour bombing. The signs might not look so amazing if the correct year WW2 started was actually written, around 3 years prior to the bombing.
I just take most of what they say and do with a rather large grain of salt.