Replicators would allow near-instant rebuilding of a city if there was political will. But the "survivor" cities we see are a mixture of old and new architecture, something that probably would not be allowed if cities were being rebuilt for "historical reenactment" reasons - if there are historical-looking buildings there, they are probably the original stuff, and WWIII survivors.
Riker made it clear that the death toll of WWIII was high, and that major cities were destroyed. But perhaps the two are one and the same thing: weapons were aimed at really
large cities, none of which would be found in North America in the mid-21st century (or even today), and/or these cities did not enjoy the advanced defenses available to First World nations.
OTOH, that a flag survives tells little about the nation that survives under the flag. And that the name of a city survives is no guarantee that the city itself retains anything physical of its past and origin.
Random points: In "WNMHDB", the magnetic storm is blamed for throwing the starship across the fearsome-looking Barrier thing, and the return journey is possible with impulse even though escaping from the storm is not (while Kirk goes there and back at warp). But the way it's presented does not actually contradict the idea that the storm took the ship across at least hundreds of lightyears. Yet to say that the storm (or whatever the crew and our heroes mistook for a storm) took care of all
the travel from Earth to the Barrier goes somewhat against the spirit of things.
That is, it seems as if the Valiant
was on course to some sort of a destination when the storm changed that course. But we know that these "storms" can be found deep inside the Sol system (VOY "One Little Step" and perhaps ST:TMP), and we know that sublight ships probed outside that system in the early 21st century already - so theoretically the Valiant
could have disappeared at Sol, the same way Ares IV did, possibly even before Ares IV. But it doesn't necessarily fit the drama all that well.
Whether "finding a 200-yr-old Earthship marker" or "finding an Earthship marker at the edge of the galaxy" was the thing Kirk considered impossible remains debatable. But what would be impossible about finding old markers? These things are meant
to be found, and apparently are technologically capable of remaining active for centuries. Surely the latter capability wasn't an accident unintended by the designers?