This is always ammusing to me because I can see how it really happened, but I can also rationalize it was faked.
America being america at that time could make me believe it was faked. We were in a race to the moon, I could see America faking it just for bragging rights and to maintain our "were number 1" status. The moon landing was incredibly important for America and Americans at the time.
Not to mention we were still fresh off the cuban missle crisis which involved Russia and the cold war. We need that victory and I could see it being faked for the good of the country quite easily.
Things were insanely easier to fake back then because we lacked the technology and communication we have today.
Shoot a rocket up, bring it back, release footage. No one really knew for sure aside from a handful of people who probably swore to secrecy as patriots doing their duty for the good of the nation.
America then isn't like today's America. Back then making America strong, feared, respected, wealthy, independent and free were politicians and citizens primary concerns. The government would do anything for the good of the nation. Now? Not so much.