[insightcheck]

I can show you one counter-example, which should work.

Hunter College High School in New York City is a public school, and its students disproportionately end up in top US universities (from: https://en.wikipedia.org/wiki/Hunter_College_High_School).

The graduates of the private high school near the neighborhood I grew up in, however, mostly ended up as accountants after attending a local public university. For specifics, you can take most private high schools in the US (even many of the very expensive ones), and they won't have as good outcomes recorded for their students as Hunter or its equivalent public schools in New York City.