Military targets are more likely. There's not much to be gained by screwing up the infrastructure of the country you want to conquer as the US discovered in Iraq. Not that I have any idea idea where the US military bases are.
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.