First, I realize this question is quite like this one, but I want to focus more on the college GPA and then relate it to two older studies.
There are a few decades old study which show that public school graduates perform better academically than do private school graduates. The newest of the following studies came in 1963.
This study compares students at Randolph-Macon College.
This study compares students at Harvard, from 1948.
The authors of this study administered a "Personal Values Inventory" test to many students and concluded that "private school students do less well because as a group they are lower in scholastic aptitude and in motivation." This is from 1963.
In the question I link to, a study is cited which notes there is no real difference in achievement after controlling for income and background. As far as I can tell, this didn't include observations like college GPA.
Are the results from the old studies I link to above reliable? Are they also persistent trends? I could not find many newer studies. Were these patterns which have died out as admissions has become more "fair." The consistency with which public school graduates did better in college might suggest that adcoms favored pedigree to the point of admitting lower quality students when they came from a private school.