The top twelve state championship leaders are as follows, no particular order.
Privates: Chatard, Cathedral, Luers, Roncalli, LCC, & Ritter
Public: Carmel, Ben Davis, Sheridan, Warren Central, & Penn. Sheridan won most of their championships in 1a without a strong presence of a private school. The other are all in the top half of enrollment of the largest class, something even the privates can't compete with.
Maybe it's just coincidence. Maybe private schools and large enrollment schools just work harder than their counterparts. But I doubt it. The evidence is right there in front of you. If you or anyone else on this forum can't see it, then there's nothing I can really do to help you.