Let's look at the timeline: Saturn V was rolled out onto the launchpad in May of '66, maiden flight in November '67. In February '66, they tested the Apollo CSM.
Meanwhile, N1 made it onto the launchpad in November '67, but wasn't tested until February '69. Even assuming Korolev's survival could have moved up the dates, there's not a lot of time here between test flights to crewed missions. The LOK wasn't even flying until December of 1970, on a Proton rocket since N1 wasn't ready. But even assuming N1 had had its first test flights in '68, how long do you think it would have taken to go from that, to testing LOK, to actually landing someone on the Moon? Probably more than a single year.
Finally, let's look at something entirely unrelated to the troubled N1 program: the matter of rendezvous. It was important enough to master that Protect Gemini was devoted entirely to getting rendezvous working as smoothly as possible. The US conducted the first rendezvous in December '65, first docking in March '66, and several more successful tests with Gemini after that (all manned missions), as well as a rendezvous in Earth orbit with Apollo 7 in October '68 and lunar orbit with Apollo 9 and 10 in March/May '69.
Meanwhile, the first rendezvous/docking by the Soviets was in October '67, by unmanned crafts that used more fuel than expected, and their follow-up attempt with Soyuz 2/3 in October '68 failed to rendezvous after running out of fuel to maneuver. Their first success with manned rendezvous wasn't until January '69.
It wasn't impossible that the Soviets could have gotten extraordinarily lucky on numerous fronts if he hadn't died, but I wouldn't call it realistic, or in fact reasonable.
The alt history takes the, unlikely due to inferior technology, path that the last ditch Soviet launch right before Apollo 11 (a launch that did in fact happen, but failed), succeeded. The show seems to make it clear that this was influenced largely by luck as the Soviets had not done any sort of "dress rehearsals" to the moon like NASA had.
For All Mankind to have a successful Soviet Moon landing, things happened differently before that launch. I thought the show would be interested in exploring what those things are, but it never was the case. As for OP, presumably they're talking about Vladimir Komarov: https://en.wikipedia.org/wiki/Vladimir_Komarov
His death during Soyuz 1, the first space casualty, could perhaps be a hinge between our universe and For All Mankind.
Alternate history is fun, but maybe best not taken too seriously.