They would still be needed for any attempted interstellar travel until the development of the Warp Engine...
...And it would be intriguing to know whether they actually were used for this before WWIII and Cochrane.
A ship that can go from Earth to Mars in a week (say, the Ares IV rescue mission) doesn't need cryosleep even for exploring the outer planets. Such a a ship could also explore the outer reaches of Sol - say, a thousand AU out - the way the Charybdis
did, with a little help from suspended animation. And interstellar missions would be quite practical with extensive cryosleep, assuming the engines didn't suffer from endurance limitations.
Khan was able to go interstellar in a pre-2018 ship, possibly a design that takes a month to go to Mars but still outperforms current spacecraft, real or postulated. It appears he wasn't supposed to be capable of that, though. But we still lack solid knowledge of when and where humans first visited a world outside Sol. Before Cochrane, or after him? Or was it him, in another of those rickety experimental craft of his?
They didn't have replicators in 2053. Or 2153, for that matter.
But our actual evidence for the survival of American cities of note comes mainly from the later centuries, where resources for hobby projects such as re-erection of ancient cities would have been bountiful.