I didn't live in a very dense part of New York; I lived on the border between Queens and Long Island, and pretty much my entire neighborhood, and all the neighborhoods for miles, basically consisted of single-family detached homes with small yards and backyards.

The main ingredient that you need for carfree kids to work is usable transit. And by usable, I mean it was frequent; if I missed my bus I would only ever be waiting 10-15 minutes for the next one. Everything I could want to go to would be reachable by bus in 40 minutes or less.

If you were to take a map of most American cities' bus networks and only showed routes that ran every 10-15 or better throughout the middle of the day on weekdays, an awful lot of them would be mostly blank. Even more if you were to require that for Saturdays as well, and pretty much all of them for the entire week. Provide the transit and people will ride; it's how Seattle managed to buck a national trend of declining bus ridership.