I made it through most of the posts before giving up and coming here to add my 2 cents, as I was a private school kid.
First of all, yeah, all of the public schools in our conference hated us because we were very competitive in all of the big sports. As far as I knew, we did not give scholarships to any sports players. We did have a program where students could work over the summer helping the maintenance staff do work around the school to help pay off tuition, but that was open to anyone.
Otherwise, we are a part of the NIC-9 (Nothern Illinois Conference 9 - now NIC-10), so we had to follow all rules for that conference, same with the rest of the public schools. People around the city know which schools tend to be good at which sports, and have the ability to choose accordingly, since Rockford has a preference-based assignment program for public schools.
Now Illinois did implement a multiplication penalty, so that all the private schools were considered to have a larger enrollment than in actuality. Reason they did this is because, over the past couple decades or whatever, the sports championships were about 50/50 between public and private schools, and they decided that was too much for the private schools to win. Now, my school was already in the highest level since we had 1200-1300 students, so it didn't exactly affect us.
Also, as a final note, although we are usually at the top of our conference, the Chicago suburb public schools still usually beat us and the rest of the schools in my conference. So the competition on the various conferences can have a fairly big effect.