In fairness to the comic, part of the problem was the TV shows reluctance to accept the Eugenics wars in the fictional timeline and intertwine them with the time travel episodes to some degree at least, even if only e.g. Rain mentioning them or something.
I don't follow your logic. If anything, that makes it even more inexplicable that the comic would portray the Eugenics Wars as a massive nuclear conflict that devastated the US.
What I mean is they're telling a story about the Eugenics war with no canon to call from. It doesn't surprise me that the writers forgot about a couple of time travel stories where characters went back in time post Eugenics but didn't reference the consequences.
Of course if Khan is exaggerating the nuclear conflict, America, as a large nation, could have had much of it infrastructure intact after the war. It depends how many nukes fell, whether they fell near any of the places we see in the time travel stories, and whether our heroes are already in an alternate timeline due to Braxton's interference.