If the moon landing was a hoax it was started by the Kennedy administration and continued under Johnson. The landing took place in 1969 when Nixon was president.
Why didn't Nixon expose the "Hoax" by his Democratic predecessors instead of continue it? It would have helped him personally and politically. And Nixon was the least-liked president by the establishment until Trump. For that matter Nixon hated the establishment as they hated him.
BTW I loathe the "establishment." I have nothing but contempt for it.
There was certainly a legitimate space program all through the 1960s, actually starting in the late '50s. No one denies that, in fact I loved everything about it growing up and accepted the moon landings as portrayed by the media until fairly recently. It was only as a lot of time passed and there were no further attempts made by any country to go to the moon that I started taking a closer look at the moon landings and the technology used and the difficulties that had to be overcome along with what the arguments being made against them were, many of which are compelling.
The timing certainly worked out well. JFK set a goal of men walking on the moon by the end of the decade and that's exactly what supposedly happened, right on schedule, to the everlasting credit of the United States if it did indeed happen. It reflected extremely positively on Nixon as he was the president when it was allegedly accomplished. He had no reason to expose anything, nothing to gain and maybe the loss of his position and possibly worse, and if it was faked he may not even have known it. We're talking about something very compartmentalized on a "need to know" basis. The intel community has often times thought even Presidents had no "need to know." In fact, many people believe JFK was offed by the intelligence community because he was making moves against the CIA following the Bay of Pigs and other incidents.
As far as mirrors, again one has to accept the establishment view at face value as to how they got there. They could have been placed there by a probe. Just because Wikipedia says otherwise is hardly conclusive evidence.