Even in cities, private schools are primarily for the affluent, and the percentages vary quite a bit by city and even within cities. larger cities also have choice programmes particularly for high school, and charter schools, which are publicly funded but not run by the local school board. In my suburb, which has a large county school board, we have magnet choices for high school, but for elementary it's your zoned school unless you need a special education programme that isn't available there.
Traditionally, parish Catholic schools were affordable and an option for working class families, but that's less true than it used to be; schools have been closing for decades. (There is a division between parish/diocesan schools, which are subsidized for members, and independent Catholic schools. Some of the independents are very expensive. Lady Gaga went to Sacred Heart, which is $50K a year.)
There are big divisions in types of private school. In parts of the South particularly, conservative Christian schools can be relatively inexpensive. Some were opened as a response to desegregation, others because of beliefs about teaching evolution in public schools and the like. Then there are expensive college preparatory schools, much more like most independent schools in the UK, and also ones with alternative educational philosophies such as Montessori and Steiner.
Religious teaching is banned in schools, and in much of the country that is actually taken quite seriously. In London I would have sent my children to a Jewish school, and paid only the religious studies fee. Here, it's fully private, and something like $15K for K-8 and more for high school. For some families, religion alone is a motivation.