Dont know how I feel about the 1 quarterfinalist proposal. It doesn't really address the issue AND takes opportunities away from private school kids to advance. Is the goal to simply reduce the private school championships OR to have an equitable tournament? Also, I don't know that it has to be across sports as I dont follow enough, but I would leave basketball as is for now (I'd actually reduce the sport to 3 classes).
I think the answer to both of your questions is yes. Did a little bit of a deep dive and here's what I came up with. This is just for the most recent seasons for what I feel are probably the 8 most popular team sports, so take it for what it's worth.
For purposes of this board we'll start with football. Private schools accounted for 11/64 teams making the quarters or just over 17%, but 5/8 classes or 62.5% being won by private schools.
For the remaining sports I did private schools who made the Super Sectionals, since it's the equivalent to making the quarters in football.
Boys basketball - 25% of super qualifiers were privates and they won 1/4 titles
Girls basketball - 28% of supers and 1/4 titles
Baseball - 25% of supers and 1/4 titles
Softball - 15.5% supers and no titles
Boys Soccer - 25% supers and no titles
Girls Soccer - 12.5% supers and 1/3 titles
Volleyball - 37.5% supers and 2/4 titles
In total private schools accounted for 62/272 quarterfinal/super sectional qualifiers or 23% and 11/34 titles or 32%.
These are just the factual numbers, interpret them and determine what the competition committee has in mind as you will.