Reading the comments here, a major part of the problem seems to be that some people move to suburbs because that is the only way to get their kids to a decent school.

Myself, I like living in a suburban area (though here in North Europe it's definitely walkable, and cyclable) but people here don't really choose between city core and suburban lower-density houses because of schools. Some love the city core, others prefer suburban areas.

(With the recent immigration and development of urban-but-not-nice satellite cities, there are a few areas that people like to avoid if they can afford it, including for reasons of school, but that's not a problem of city cores).

So, it looks like a major step in American urban planning would be to fix failing schools.