Further to what Mike says, I'm not too sure on what percentages are re middle class families children going to private schools is or in fact what is called middle class.
If my experience is anything to go by, it'll vary a whole lot and it was my background if anything that had two girls of mine successfully going through the public system.
My parents were not wealthy and separated when I was young but regardless, because of their upbringing, a private school was not going to be on the agenda without a scholarship, and all siblings got educated well enough even if I did my engineering as a part time student.
So even though I was earning a heap more than my parents without be considered well off, but substantially middle class, it was still public schooling for the daughters, both doing professional courses.

As Mike says, it'll depend a lot on the area you live in and how supportive parents want to be with the school, but there are so many factors influencing education in this day and age, sometimes it'll be the students attitudes and that of teachers and parents that could have an effect more so than what facilities a school has.

I certainly would not fear that our public education system is anything to be alarmed about, and it still handles a greater percentage of students.