As long as private schools pick and choose who attends the schools, any notion they are better is just hogwash.
Not to mention these schools suffer from high turnover of teachers because the pay and benefits are so bad in those schools, especially the parochial ones.
I bet if I walked into one of these "elite" private schools that are fairly common in the East, I bet they wouldn't be all that much different than public schools in terms of curriculum. However, there would be lots and lots of "high-achieving" kids in those classes.
Article talking about the above-linked book is here.
You suggest that public schools are open to reform and new educational theories in a way that private schools are not. Why do you think that is?
STL: It appears that there is a danger in the autonomy that private schools have. The teachers aren’t required to be certified, there is less professional development happening, they’re not held accountable to the same kinds of state curriculum standards and tests. And so when we look at scores on those things it just makes sense that the schools who are hiring teachers who are certified and have been educated in a way that helps them understand all the current educational reforms and the research on learning—that those teachers would be more effective. Particularly more effective at educating students on the state standards...So, yeah, the autonomy of private schools may not be all it’s cracked up to be.