Good post. In addition to discipline I would add the ability to pray and regular school mass. There is an effort however to limit the masses offered to students out of deference to those students who attend but are not Catholic and might be offended. You hit on something important which is the number of lay people as administrators in Catholic Schools, historically when this happens Catholic Schools become prep schools only interested in accumulating many dollars to pay teachers and build new buildings.
Some years ago, needing more revenue, the school faced the question of admitting non-Catholic students. Ultimately the issue was finessed in favor of admitting such students with certain understandings: Catholic students are required to respect the faith commitments of non-Catholic students and vice versa; all students are to attend a Tridentine Mass (old Latin rite) on each First Friday of the school year and to attend daily rosary. Those last two requirements do not require non-Catholics to participate either in Mass or rosary but the school has no surplus personnel to supervise them during Mass or rosary. Non-Catholic parents have yet to object to the school's decisions in these respects. When it comes to quality education, Catholics and non-Catholics can work together for the best interests of their children and society.
The school is administered by lay people but they are not chosen on the basis of incestuous "edjumakashun" degrees but rather on the basis of their commitment to the educational and faith commitments of the school. As a result, though the school does not cherry pick its students and has graduated a few kids with substantial disabilities of intellect or psychology, its students are two years or so ahead of parochial as well as public schools.
A sampling of why: Saxon Math, Fr. Henle's WW II era Latin texts (starting in later stages of grammar school), Samuel Eliot Morison's Oxford History of the American People supplemented by Schweikart's Patriot's History of the American People (the former having been the standard text at Harvard for many years), world history by former Yale Academic Dean Donald Kagan and his associates. Rigorous requirement of grammar and syntax without vulgarity, not just as courses but also on all papers and examinations and oral classroom responses. Quality texts for quality education. A Catholic culture open to admission of other students, Catholic moral standards shared by most conservative Christians, an understanding of American exceptionalism, a commitment to quality education.
The school does administer qualification examinations for incoming students in order to determine what grade they are ready for. Incoming kids (including many from parochial schools) must choose between catching up in summer school before admission or starting at an earlier grade level than they had attained in other schools. Once they find their level, they tend to do well. Each kid has only one chance at an education and it is more important by far to get that right than to do it quickly. There is no notable hurry that justifies second rate education or second rate morality.
There are numerous schools like this (independent of one another) in Illinois and elsewhere. To the extent that they grow in numbers and geographical spread, along with similar Reformed Christian academies and Chassidic Jewish academies, our nation, our civilization and, most importantly, our God will be served in the future. Keep such efforts in your prayers.
God bless you and yours!