But I agree, I think it's fair to say that Khan's description of 'chaos and barbarism' from 'coast to coast' is not reflected in any of the time travel shows post 1992/ pre-mid 2100 we saw, albeit that those travellers were being plopped into a parallel past according to many worlds theory. It's possible that the writers conflated the Eugenics Wars and WWIII.
Still, Star Trek is set in a fictional world so any assumptions about how much fictional information they have from any particular era is purely speculative. Having said that, if Bride of Chaotica survived the nuclear strike, it could not have been that bad. Maybe Khan's definition of 'chaos and barbarism' was because the Prada and Guccii stores and Fox network were bombed out.
I suspect such to be the case.
Given the references to Bride of Chaotica, the situation post-Eugenics Wars in places as diverse as Los Angeles, Detroit, and San Francisco as seen across the various series, the fact that we know of Buck Bokai and why he mattered to professional baseball in North America, the survival of hockey as a sport...the Eugenics Wars had to have been public but relatively self-restrained compared to WW 3.
Also, the idea of all of Canada as part of the "Great North Wasteland"?