I wouldn't say that it's quite that simple, but it rings true if you think about it from the perspective of the de facto power groups.
When private schools exist, those power groups send their children to the best private schools, and few "important" people care about the public system.
However, when private schools do not exist, and those power groups have to send their children into the public system, then suddenly a whole lot of "important" people will care about the state of the public system, which is to everybody's benefit. (Also to the benefit of those "important" people, because they also benefit from the positive externalities of a well-educated society.)
Obviously, it is possible for a public school system to degenerate into a de facto-tiered system, where schools in poorer districts fare much worse than schools in rich districts. So this is just one factor among many, but it is a factor.