Yes, amongst the Americans I know, I lived in SF for a while, the suburban middle classes are far more likely to have their children in public schools, it is very rare that they send them to private schools, unless there is some sort of family tradition, and the public schools they send them to are regarded as providing a homogenously good standard of education. Teachers I know who have taught expat kids comment that American pupils who have been through the American public system have a better level of basic literacy and maths than those who have been through the British system (state or private) but are not as advanced in some areas eg literary criticism. People simply do not feel the need to go private.
The problem with the US system is that a deprived area in an American city will rarely have an infrastructure, not just education but for all public
services, that could be described as better than third world. It is quite shocking to stray into such an area in SF or LA, roads are not repaired, street lighting will be bad, public hospitals appalling and as was highlighted in the New Orleans floods, they cannot even rely on equal access to emergency services. The schools are dire. There is no way out of deprivation on that level, and no will in either party to do very much about it, look at Obama's problems with healthcare reform.
However what they do have in the US is a greater culture of philanthropy. It is far more likely that the average wealthy person in America will be paying out to charity, especially their alma mater.