The problem demographic is couples with kids. Our urban schools are bad, so they leave the city to find good schools. But people with no kids -- either before children or after they leave for college -- are living in cities.
Good urban schools exist, but there aren't enough of them, hence there is intense competition to get into good urban schools. To your point, this then ends up driving more families away from dense urban centers.
It's not just that — due to desegregation (itself a good thing!) one can't ensure that one's kids will go to either a good school or one nearby. Parents want the best for their kids, and will do their best to ensure their kids' success (note that Mr. Obama's kids went to Sidwell Friends, not to a public DC school).
High housing costs get even worse when you are paying for 5 people instead of 1 or 2. This multiplication factor also carries over to many other costs so a city block with artisanal restaurants and locally-grown grocery stores just doesn't cut it for family budgets.