As often stated, it would be easier to actually go to the moon, then to maintain the silence of the thousands of people who would be in on the conspiracy.

The Manhattan Project. Tens of thousands silent for years, not a peep. Threatening people works.

Also, if the US didn't go, does anyone really think the USSR wouldn't have called them on it? They were tracking it all too.

The USSR was in a famine and traded their silence for US grain supplies.

I suggest watching a documentary like American Moon, all the counterarguments have been comprehensively addressed.