In fact, as I recall S1, For All Mankind is basically premised on the notion that the N1 worked and therefore, the Soviets kept their early lead and landed on the moon first.
Soviets were the first to land on another planet orbiting the sun, Venus.
While the moon orbits earth.
For example, coming back to 75 tons N1 and dealing with more dockings on orbit. That would at least reduce number of engines on the first stage; maybe not a very good idea though.
Or coming back to another idea, having N2 (which is N1 without first stage) a priority and fly to the Moon with ideas his team had before (in 50-s? there was a plan with multiple dockings and refuelings, which would use more modest rockets, but N2 with ~25 tons of payload would certainly help). N2 was even in a better tested state - it was the first stage of N1 which was the most problematic.
Or making even better with Chelomei (Korolev approved project Zond, which was a kind of joined project between two design bureaus) and use Proton for multi-launch schemas. Plenty of opportunities.
If Korolev would be lucky enough to avoid Komarov's death, Russians would have some spare time - in addition to momentum - to move things forward.