unmanned missions to the outer planets, etc, etc, etc.
Exactly why send a (wo)man when a robot can do the job without having to worry about life support?
As an aside to the USSR v US space race, a very large portion of the time the Apollo missions were enroute to the Moon they required telemetry for guidance, as you know the earth rotates and this data was provided by Australia and.....the USSR...if the USSR could have scored a point regarding the faking of the moon landings this was the perfect point at which to do it.
The conspiracy theory didn't start until after 1974 when the film Capricorn One came out, that was a film regarding the faking of a mission to Mars, and then the theory gathered pace with the invention of the internet as with all good hoaxes/conspiracies etc.