make. By my quick count, there are only 11 private schools out of 68 total in the 5 major conferences. Of those 11, 3 (ND, USC, and Miami) have or have had elite football programs in the last 40 years, and 2 (Duke and Syracuse) have consistently competed for basketball NCs in the last 20 years. Percentage wise, I think that probably compares pretty favorably with the state footballs. Heck, if you look at college football, ND, Miami, and USC have 7 national titles and probably as many runner up finishes dating back to 1980. You could almost argue the private schools are disproportionately successful in college football.
The number of non-major conference football schools in the upper echelon of college basketball probably has decreased in the last 20+ years, but that's not relevant to ND.
EDIT: I now count 12 privates out of 68 - USC, Stanford, TCU, Baylor, Vandy, Northwestern, Syracuse, Duke, Wake, Miami, BC, and ND.