I'd be interested in work in a any country / region

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This isn't specifically about private schools, but there's a book called The Case Against Education that argues that the education system in general adds very little in terms of human capital, and is mostly a way of certifying intelligence, conscientiousness, and conformity.

Maybe not quite what you are looking for but Lant Pritchett's work on schooling such as his paper "Where has all the education gone?" looks at wherever shocking in general will add value to society vs only change who attains socioeconomic success?