How about just a good and reasonably fast commuter-rail network? It's faster to get to many suburbs of Copenhagen by public transit than to drive, because we have: 1) extensive commuter rail; and 2) not many freeways.
I don't think it's only that American cities aren't actively discouraging driving (e.g. by banning cars from city centers), but that they are spending huge piles of money actively encouraging it, by building 10-lane freeways and the like. If you build rail and not freeways to all the suburbs, most people take rail for routine trips.
America is big. The population is very spread out. It is normal for an American city to have 1/20th the population density of Copenhagen. Commuter rail would just dump you in a sprawling suburn with no economically viable public transportation. Autonomous taxis will make it workable, but that is a decade or two in the future.
I don't think it's only that American cities aren't actively discouraging driving (e.g. by banning cars from city centers), but that they are spending huge piles of money actively encouraging it, by building 10-lane freeways and the like. If you build rail and not freeways to all the suburbs, most people take rail for routine trips.