Again, we cannot ignore the effects of legacies. Private prep schools serve the same demographic as the “elite” private colleges - the wealthy and powerful. That means that the percent of private prep schools students who are legacies in “elite” private colleges is extremely high. This alone would triple or more the percent of private high school students who are accepted to “elite” private colleges.
Add to that the advantages in personal treatment by GCs, the personal connections that GCs have to AOs at “elite” private colleges, and the fact that the AOs at the “elite” private colleges are often graduates of the same private prep schools, or similar ones, and are therefore more inclined to prefer graduates of high schools which are like the one they attended.
Time was that places like Yale or Harvard always made sure on a relatively high percent of their incoming class was from a number of private prep schools. While this is likely no longer explicit, I wouldn’t be surprised if they maintained this indirectly, by legacy preference, preference for athletes in sports that are common at prep schools, but not at private schools, etc.
Only part of this works to the benefit of students who are not wealthy and/or legacies, but some of those practices do, such as personal attention by counselors.
If one wants to know how legacy and wealth play a very large part in admissions from private prep schools, one just has to compare the number of admissions to “elite” private colleges from private prep schools to the number accepted from the top magnet schools. As a rule, the relative number of students accepted from a private prep school is triple or more the number accepted from an equivalent, or better public magnet school.
For a student who is not wealthy and/or a legacy, there is likely little benefit in admissions in attending a private prep school, compared to a public magnet school, all things being equal.