Due to the potential military implications, one major part of the Cold War was the Space Race. The Soviet Union claimed to have the first orbiting satellite, animal in space, man in space, woman in space, space walk, and Lunar rover.
https://www.history.com/news/from-sputnik-to-spacewalking-7-soviet-space-firsts
How could the US claim to have been victorious in the Space Race with anything less than putting men on the moon then returning them safely?
Which is why the Soviets would have yelled “bullshyt” if it was faked