I went to private schools my entire life...the students there don't want to be taught any more than their public counterparts.

So did I (However, my school district's gifted student program was held in several public schools over the years I participated). My wife attended public schools in 2 adjoining school systems over her career. One of my SOs when I was in high school went to the public high school in my district. My impression, which I base only on interaction with other students inside and outside class settings and on discussions I had after I got to college, was that the education you got was what you made of it. Going to a "better" school was neither necessary nor sufficient to learn more. It also wasn't necessarily indicative of the level of support being offered at home.