They did try, but the N1 moon rocket was way too ahead of its time. It was powered by 30 small but more efficient NK-15 rocket engines, and the computer controller at the time was not efficient enough to respond to that many rocket engines at the same time, and had led to disastrous consequences.

The Americans simply built the Saturn rocket with 5 very powerful but less efficient gas generator F-1 engines, and got lucky. Technically it was impressive, but conceptually it didn’t break new grounds. There is a reason why there hasn’t been any manned mission to the moon in 50 years since 1972, because this kind of rocket is already out-dated and no engineers can or know how to reproduce these old designs anymore.

To add insult to injury, SpaceX’s Starship (probably the only one in America making significant progress on space rockets these days) based its design on the Soviet N1 moon rocket, powered by 33 small but efficient Raptor engines, which vindicates the Soviet design. Even 55 years later, SpaceX with all its modern technology and computers are still having trouble getting the Starship to fire up properly. They’ll probably succeed soon but for now, nobody has been able to get the concept working.

On the left, the Soviet N1. On the right, SpaceX Super Heavy

I dunno, probably didn't see the point. They were busy making probes to send to mars and Venus.

Another thing: when the N1 program flopped the Soviets still didn't have a moon lander, a bunch of space rendezvous experience, et cetera. The American moon landing was really accomplished by two programs - Gemini and Apollo - while the Soviets tried to do everything at once and the result was the largest non-nuclear explosion of all time.

So they decided to just focus on what they had an advantage in: automation and precision, which left the moon program on the wayside.