Now we tend to self segregate and choose to live in places with people like us. As professionals choose to live in cities instead of suburbs, schools around them will likely be better funded.
1) Curriculum 2) Safety
It's one thing to make the argument that most public schools are mediocre and the parents need to be more involved in their kids lives to make sure they stay focused and do well. But it's quite another to willingly send your kids to a school with drugs, gang violence, and an abundance of teen pregnancies just because you don't want to commute to work every day.
For example, in NYC the center has gentrified; most of the slummy areas of Manhattan are now desirable to live in. While areas that used to be quiet and suburban in Queens and the Bronx have ended up being sketchy, in a sort of inversion. In most of Europe that's also happened; if you can help it, you don't want to send your kid to a school district in the Paris suburbs, if there's any way you can manage to get them into a good urban district.
As a different example, Houston is developing into a strange multiple-rings shape. The city center is gentrifying (medical center, Rice area, downtown), while the inner suburbs are going downhill as the housing stock gets older (60s/70s-construction suburban neighborhoods aren't looking so good anymore), and then really far away suburbs with newer construction (20+ miles, 1990s-present housing stock) are upscale again.
It seems that what has happened is that, for the majority of American cities, they grew as an either/or option: you're either in the city, or you're in the suburbs, and the car enabled that in cities where it could. The cities which were geographically constrained, such as NYC or SF, were unable to sprawl outwards, and have commutes within reason, so the middle-classes stayed. That means things like good schools are able to exist because the demand was always there from the middle classes that didn't leave.
Part of that I think is also because some of the costs that incurs aren't properly charged. There was a proposal in Houston to charge developers for the cost of new infrastructure. For every 10,000 new homes built in a new suburb, for example, you can estimate what new capital infrastructure they'll need: on average 1 elementary school, 0.5 high schools, 1 fire station, 1 police station, X miles of roads, X miles of sewers, etc. (whatever the real numbers are). Add the estimated cost of that up, divide by 10,000, and charge that as a per-house assessment on new developments. That proposal failed, though, so new developments' indirect capital costs are subsidized by homeowners in the older areas of the city.
The interesting component is how the development gets financed in a Ponzi-esque scheme. Suburbs are built out with municipal bonds, which subsidize the build outs of services (schools, police, fire stations, roads) while keeping property taxes low. Once the suburb is completely built out, the taxes double or triple to pay back the bonds. Then the next suburb just north or south of it is built with more new bonds and the cycle continues.
There's a stripe running east west through the center, following the hills and the area immediately south of the hills, from Santa Monica to Griffith Park. That stripe is notably higher priced, with prices within the stripe generally increasing towards the ocean.
Then there's a ring around that stripe where prices decrease for a while, and then move up again to be sort of traditionally suburban in pricing.
And then there's the strange outer regions like Topanga / Calabasas / Malibu, where you move waaaay up the scale again.