As for pricey private schools outperforming public schools:
Academic League
The Bishop’s Academic League team participates in the Northern division of the City League. Recent results include winning the Northern Division championships in 2011 and 2013. The Bishop’s School has lost, though, both city championships to La Jolla High School in 2011 and Madison High School in 2013. The team also competes in NAQT tournaments throughout the year, finishing 2013 with an invitation to the National Finals of NAQT.
FWIW, here are the results of the Academic Decathalon. It’s open to all public and private high schools. We’d need to know the ratio of public-to-private schools to get an accurate picture of representation at top levels, but I do know that many prestigious private schools compete in this competition. Look at who the winners are, check the history over the years, and you’ll see that public schools are at the top in almost every single event over the years:
