IIRC, with the N-1 program's budget limitations (as well as time pressure) there was neither time nor funding available to do the kinds of comprehensive tests of the engines and other systems that were done in the Saturn program. The N-1 had huge numbers of engines because the individual engines were far less powerful than the American F-1 rocket engine, so I would hope at least that the smaller Soviet engines were static tested--but possibly even that step was skipped? In any case, the Saturn program included static firing of an ensemble of 5 engines arranged as they would be on the first stage; the analogous project in Russia would mean assembling an array of 40 engines in the same confines the N-1 booster stage would have, which is clearly a gargantuan project!
However if they had had time and resources to include such testing in the plan, I daresay the N-1 might have been made to work well enough. It means that it will not do to say that if the plan is OK'd in 1961 instead of '64 that it will be three years ahead; some, perhaps all, of that time would be needed to do the extra steps. But if that were done, then at the end of it there might be a flyable rocket design after all, one that does not destroy the launching pad and does succeed in launching heavy payloads into Earth orbit.
From what I understand of the Soviet organizational culture and the role space exploration played in Kremlin politics, I doubt one could project this outcome without a POD going back perhaps as far as Stalin's death or even before, to make the commitment to following through on space goals and progressive advances there stronger and to establish a stronger space organization instead of the competitive and mercurial rivalries and fads of OTL.
It's actually hard for me to imagine a more staunch supporter of a space effort emerging to take power in the Soviet system than OTL Nikita Khrushchev. And it is hard to see how he could avoid his fall from power in the mid-60s, which threw what organization there was into disarray. Perhaps if he had hit on more successful economic reforms and avoided the Cuban Missile Crisis, he might have hung on and had more resources to funnel Sergei Korolev's way--and perhaps the Chief Designer might not have died on the operating table too. But all of this is long shots--and if Khrushchev had not precipitated the Missile Crisis American-Soviet relations might have been more strained. Ironically the manner in which Kennedy and Khrushchev got together OTL to talk each other down from the crisis was crucial in opening better communications between them! I suppose though that if the Soviets had never attempted to put missiles in Cuba in the first place, the two leaders might, over a longer time (especially if no Missiles of October somehow butterflies the Kennedy assassination) find their way to a detente anyway, one that would not be disrupted by leadership changes on both sides. (I still think Kennedy would have wound up as bogged down in Vietnam as Johnson did). But even avoiding the Caribbean mess and with the Soviet economy performing somewhat better, that doesn't much improve the situation in the USSR circa 1961, nor is there much prospect of a less fratricidal, more cooperative and better organized Soviet space effort. Some of the conflict was due to personality clashes and ambitious empire-building; but other aspects of it boiled down to serious fundamental issues. The Soviet military favored development of hypergolic "storable" fuel based rockets such as the Proton as these fuels were more sensible for their missile programs; this weakened Korolev's position as he tried to avoid them in favor of kerosene-liquid oxygen systems. IMHO I'm on Korolev's side here, for space launches, but the advantages his rivals who were more willing than he to go with the same sorts of engines the military wanted anyway enjoyed should be obvious. Even if an ATL USSR would be richer than OTL come 1970, that wealth was not yet evident in 1961 and so a decision by top leadership to devote extra resources to a space program that in no way would yield any direct strategic benefits seems doubtful. Nor is there any guarantee that even if he had those extra resources and an early start relieved him of time pressure, that Korolev would necessarily have used them wisely and performed the necessary tests. And to be sure, though it might have been made workable, the N-1 design was certainly inherently risky--bigger than a Saturn V, yet less capable.
I've wondered elsewhere whether Korolev might have gotten somewhere with an incremental improvement on the R-7 rockets than OTL the early Soviet space program and to this day the entire Soviet/Russian manned program has relied on. Essentially the same rocket that launched Sputnik I is today carrying cosmonauts and Western astronauts to the ISS! Though current designs are indeed a stretch of that old rocket, intended to be an ICBM, I wonder why there weren't more ambitious increments of it earlier. Say the cluster of booster engines at the bottom were increased from 4 to 6, for a total of 7 engines, and the stages stretched accordingly--might the expanded rocket be a more suitable workhorse for LEO missions, including multiple launch assembly of a moonship there? If we can imagine the N-1 being nixed completely but the funds allocated for it being shifted over to this improved rocket and to Soyuz development, might we not see more accomplishments, sooner?
I tried to work out a possible mission profile, involving gradual accumulation of hypergolic fuel tanks and engines in LEO, and a final double launch of the manned elements plus a hydrogen-oxygen Lunar insertion stage, for an EOR-LOR mission to land a single cosmonaut on the moon then quickly return him to Lunar orbit.
