There is one reason I could believe it was fake. And that’s because Kennedy had vowed to send a man to the moon and back BEFORE the decade ended.
As for Sputnik, I remember as a kid looking up at the sky at night and seeing the little white dot that was Sputnik. It was scary as hell.
I recall looking at some site years ago with all of the supposed photos from a stage somewhere - seemed the argument was with lighting, shadows, etc.
I wonder if the government DID stage some photos. Either to show if the landing(s) did fail and to use as propoganda, or as sort of higher quality PR photos. Of course, that is assuming that the photos that were being discussed were fake. I imagine that lighting on the moon (with a bright sun and perhaps a bright earth reflection) is different than here on Earth.