There are two other things that differ from the N-1 vs modern rockets
Poor on-board computers
The Apollo 11 mission, for instance, had its famous 1202 alarm
Through exhaustive testing, the team at MIT’s Instrumentation Lab designed the computer such that it would never be full at any point in a mission. There would always be space available for the next program, rules in place to interrupt a program if something needed to be run immediately, or space to schedule the program after whatever was currently being run through the computer. But when Apollo 11 was descending towards the lunar surface, the computer ran out of Core Sets. This is where the 1201 and 1202 program alarms come in.
In other words, the Apollo computers (which were bleeding edge at the time) just couldn't process everything that was going on. Whatever device you are reading this post on not only has vastly more processing power, it's also built on well-known and tested technologies.
The N-1 rocket had KORD as its central control computer and... let's just say that the Soviet computer asked the Apollo computers to hold its beer while it tried something
The KORD was found to have a number of serious design flaws and poorly programmed logic. One unforeseen flaw was that its operating frequency, 1000 Hz, happened to perfectly coincide with vibration generated by the propulsion system, and the shutdown of Engine #12 at liftoff was believed to have been caused by pyrotechnic devices opening a valve, which produced a high-frequency oscillation that went into adjacent wiring and was assumed by the KORD to be an overspeed condition in the engine's turbopump. The wiring in Engine #12 was believed to be particularly vulnerable to this effect due to its length; however, other engines had similar wiring and were unaffected. Also, the system's operating voltage increased to 25V instead of the nominal 15V. The control wiring was relocated and coated with asbestos for fireproofing and the operating frequency changed. The launch escape system was activated and did its job properly, saving the mockup of the spacecraft. All subsequent flights had freon fire extinguishers installed next to every engine. According to Sergei Afanasiev, the logic of the command to shut down the entire cluster of 30 engines in Block A was incorrect in that instance, as the subsequent investigation revealed.
In other words, KORD got confused and decided to turn the rocket off and lock it all together. Whoops! At least the empty crew capsule returned safely... KORD wouldn't do as poorly in subsequent launches, but mostly because there were other things (like exploding engines) that were beyond the control of KORD.
SpaceX computers have not really been the cause of any mission failures that I know of (premature shutdown on Falcon 9 flight #4 might have been human error). Indeed, SpaceX rockets are capable of doing something no 1960's rocket could dream of: landing again for reuse later. SpaceX even has their capsules dock automatically with the ISS. SpaceX is now regularly using Falcon 9 rockets, with 9 engines. Coordinating more engines should not be significantly more difficult.
The Soviets were in a race
Remember that N1-3L was launched on Feb 21, 1969. By that point, the Apollo Program had already sent and returned Apollo 8 from the Moon, and Apollo 9 was just over a week away. N1-3L would be roughly equivalent to Apollo 6 (last uncrewed Apollo mission). N1-5L was then "launched" on July 3, 1969 (it didn't make it very far), around two weeks before Apollo 11 launched.
SpaceX is not in a race here to get Starship working. Indeed, they have been taking their time with the fully-stacked Starship. The only real competitor it faces is NASA's Space Launch System, which took nearly a decade to get one rocket off the ground, and is (for now) exclusive to the Artemis program (21st century Apollo redux). The SLS is also far more expensive and takes a long time to build, as none of it can be reused.