The populace has always been gullible but, during the 1960’s, a misplaced trust in government made the people ripe for fleecing.
The CIA was conducting unsanctioned experiments on unsuspecting civilians. The Army was developing and/or deploying biological and chemical weapons. The Navy amplified/invented the Gulf of Tonkin incidents which the president and Congress used to justify escalation of the military engagement in Vietnam. The CIA was trying to overthrow every foreign government that would not yield to American influence. The Government – at a minimum – obfuscated the facts of the JFK assassination (to this day) — and those are just the things that are public record.
Given the decade’s ‘long train of abuses and usurpations’, is it so much of a stretch to think the government would fake the moon landings to win a race that it had been losing since 1957?