Sooner or later it becomes an unmanageable mess.
That's likely taking it too far, while there are occasional contradictions, with a little thought the majority of Star Trek makes sense.
The Eugenics/WW III concept becomes a mess in itself when you pay close attention to it. Even when you don't pay close attention you can catch the problems.
A lot of it is due to ret-conning every time a new twist is put on the subject.
Spock says the
last of earth's world wars occurred in the mid 90's.
Ds9 says the Eugenics wars happened in the 22nd century. TNG says WWIII happened around 2053.
Yet Colonel Green led a genocidal war in the early 21st century.
Part of the problem as pushthebuton mentioned is adding too many ideas to the idea until it starts contradicting each other.
I've always assumed that the eugenics war would have been concentrated in the parts of the world where the "supermen" gained power, and not involving America. We actually don't get involved in every war.
From 1998 through 2003 there was a major multi-country war in Africa that killed five and a half million people, most Americans never even heard of it. Easy to believe that there could be a major war ongoing elsewhere in the world and it wouldn't directly impact the average citizen in Los Angeles.
I agree, it could have been just centered around that area.
But when Voyager went back in time to 1996 and orbited earth, the Eugenics Wars (or it's aftermath) would would have been one of the first things they would have picked up.
Once on the surface, no one seems concerned that the other half of the world is bombing itself out of existence--life seems pretty quiet and normal.
I don't recall it being said the Eugenics Wars (plural) went nuclear. Since they do use the plural, its possible that there were several wars going on against the Augments worldwide. The one that took down Khan was the last one to end.
With the post atomic horror along, with the colonel green wanting to euthanize civilians for radiation poisoning, it suggests the war was nuclear.
Spock said in the 90's earth was on the verge of the dark ages.
But that depends if you count the Eugenics war and WW III as the same thing, which TOS pretty much said outright.
The point is, in a funny way, Trek is like fundamental religion in the way it piles on the horror and destruction for humanity before the 'salvation' comes.
The Eugenics Wars, Sanctuary Districts, Earthquakes, World War III, Colonel Green, genetically engineered superman Singh, The Post Atomic Horror-- super overkill.
I think the original idea was that there was a world wide war that nearly wiped out humanity and civilization and it scared them enough to began working together and change for the better.
It was simple and effective enough, but it got inflated later with each new story added to it.