Talk about hyperbole.
The moon mission has been prepared for before that speech took place. It wasn’t just starting from scratch and hitting the moon in 7 years, instead the speech was more public disclosure of a deadline that looked achievable but would hit after his presidency (1960 + 8 being less than before the decade is out: 1970).
The biggest problem is we already did the easy stuff. Playing tag with the moon is unbelievably easier than a permanent moon base or landing on Mars and getting back to earth.