Soviet moon mission had real trouble with the booster, which had many engines; this is not something you can realistically control by hand. In short, the boosters kept blowing up: https://en.m.wikipedia.org/wiki/N1_(rocket)
(That setup was later avenged for all its tragic mishaps by SpaceX's Falcons.)
It's not as much the number of engines, which troubled N-1. For example, when Soyuz launcher starts, it has 32 chambers - not engines - running simultaneously, and it's a rather robust and reliable rocket.
What was the probable reason is the lack of testing of the huge system of the first stage of N-1 all together - test stands required lots of money and also time, both were in short supply (N-1 is surprisingly a rather inexpensive rocket, in comparison, say, to a better flown Energiya). Engines on the first stage were also untested, by design - they were single-start engines, so they couldn't be tested on a test stand before actual launch.
Both of these problems were taken into account when Energiya was developed. It had a huge "stand-start", both for testing and for launching, and engines were fully reusable, with the idea of later returning the first stage of the launcher (horizontal landing, with parachutes) and flying it again.
That's one of the options they had for moon flight, and arguably the critical part that closed their chance to get there first.
But a lot of other options, doable earlier, would have been possible if there was less perfectionism applied to automated flight - that's the argument I've heard.
N.B. Currently used Soyuz spacecraft (not the carrier rocket) is derived from the moon program capsule.
Space booster stage flying on Proton and Zenith, Blok D, is also derived from the fifth stage in the Moon expedition stack. Its engine is also the predecessor of the main engine of Buran. Lunar spacecraft technology helped with later creation of space probes, like Vega, and flying today space booster Fregat.
No, it was a great messup with Kremlin messing up the space programme big time, appointing talking heads, and "micromanaging the launch proceedures all the way from Moscow"
(That setup was later avenged for all its tragic mishaps by SpaceX's Falcons.)