First of all, that's a bit misleading. There is only one 6A private school in boys basketball, and there aren't any in 6A girls basketball. There are no 5A private schools in boys basketball, and there are only two girls 5A private schools. There is one 4A private school in boys basketball, and there is also one girls 4A private school. So you wouldn't expect many private schools in the big classification games.
HOWEVER, I'm not sure where you are getting your 20-4 split from. It's clearly not correct. In boys, you have 3A Lincoln Park versus North Catholic, so that's two. In 2A you have OLSH versus Serra, so we are already at four. And in 1A you have Vincentian versus Nazareth Prep, so that's already six and we haven't gotten to the girls yet.
In the girls you have North Catholic in the 4A game, Shady Side in the 3A game and OLSH in the 2A game. That's three more, for a total of 9 private schools, not 4.