For those outside of Cincinnati; Moeller, Elder and LaSalle are "Diocesan" schools, which means they are controlled (in a very loose sense of the word) by the Archbishop and are "open" schools in terms of enrollment. St X is a private school run by the Society of Jesuits and is subject to very little control by the Archbishop.
Again, I loved my years at Moeller and I am still in close contact with many of the faculty, but here are some things to consider: tuition at Moeller is over $7,000 per year, added to that is the cost of a laptop computer which can be paid upfront or amortized over the 4 years a student will be at Moeller. Those facts alone make the school completely different from the public schools.
In addition, the school receives state funds in the form of Auxiliary Services something on the order of $700 per student per year, funds for fire drills, tornado drills, attendance, grades, and other functions teachers perform along with transportation of our students to and from school by public school district busses. That's a huge chunk of state public education monies that show up in the "per student" costs of public schools, but are not reflected in the "per student" costs of Catholic schools.
Unlike Sycamore High School, just a mile or two up Montgomery Rd. from us, we could (and often did) dismiss students for their behavior. In some cases those same kids ended up at Purcell - Marian or McNicholas, but in most those kids usually ended up in the local public school. Their behavior and their academic problems became "public school problems."
I want to add that the school did not dismiss students lightly... you push a couple of $7 grand tuitions out the door and you have to start looking at your budget. You loose five kids and you're looking at a faculty position.
Still, if a kid smoked dope on campus, or got ugly with a teacher, he was gone and matriculating at Sycamore/Madeira/Reading/Indian Hills/Mason/Fairfield... you get the idea... the next day. That is a huge difference... especially to the faculty and their sense of control of the classroom.
There was a spiritual presence at Moeller. We could bring religious education into the academic mix and we could work freely with the Church's spiritual guidance programs to help students and families in ways which public schools cannot. Again an advantage over public schools.
My last year at Moeller was 99 - 2000. At that time we were not an open access school... and could not be required to be one... therefore we had no "challenged" students. We offered exactly NO vocational education programs and therefore did not have to serve vocational students. Kids had to take an entrance test and we openly discouraged kids with poor test scores and poor elementary school grades to look elsewhere. We did not have any LD teachers/tutors on staff and we did not honor any education support plans (which all public schools must serve). We did not have to honor unfunded mandates from the state for proficiency tests (though we did eventually take the tests), ESL programs, or provide support for challenged students nor did we have to "mainstream" students. Our AD funds supported exactly zero, zip, nada athletic programs for female students, Title IX? What Title IX? Hell, we didn't even have to deal with female students unless we chose to do so.
Now, despite all of those advantages we never surpassed Sycamore, Indian Hills, Walnut Hills, or Madeira in the number of Merit Scholars we turned out.
My point is not that Catholic education isn't worth the investment, but that a comparison of any two systems has to assume that both operate under similar conditions. They clearly do not.