Those family members who have done both public and private schools say the best students are in private school. The reason isn't that public school fails these kids, but because of the differences in the people in private school. Private school parents are highly motivated and expect their children to be. The teachers, administrators, and parents are usually all on the same page in terms of discipline, course work, ethics and expectations.. Most private schools are smaller, and don't have the range of socio-economic groups, children with disabilities, or language issues that public schools have. Parents have a monetary investment that makes them demand success from their children. There are no un-involved parent in private school. And private schools have more discretion when issues do come up.
I'll just say, having spent 6 years in private schools and 5 in public schools (good ones without major social, ecomomic, or practial issues), I think private schools were HUGELY better, but I disagree with your generalizations about why.
for context:
grade 1-6 I attended 5 different private schools in 4 different states
grade 7-9 I attended public junior high
grade 10-11 I attended public HS (and graduated at end of my junior year).
in the public schools, I was there on scholarship, along with some other kids, although more were there on their parent's money. so we had varied social and economic backgournds. we were low-middle class (that is, foodstamps, but not project housing).
we didn't have much racial diversity in either the private or the public schools... but that was reflective of the communities where the schools were. my parents were high-achievers but low involvement with us, and almost no teaching/homework/etc involvement once we started school. we didn't live in a disciplined household, chaos was essentially the rule. expectations were high, but support was absent. private schools were smaller, but class sizes were more or less the same throughout. my parents were equally uninvolved when I was in private and public school. nonetheless, I graduated at age 15 from HS with a 4.0. my parents had no monitary investment in my private schooling because I earned it with scholarships.
some of my classmates were *very* wealthy, and some of those kids had equally little contact and support with their parents (although more with their household staff), although their parents certainly did have money committed.
here's the big differences I see from my own experience.
the private schools I went to taught thinking. they taught useful, flexible, portable mental skills. the public schools I went to taught information. dates and facts. memory work. the private schools taught how to learn, how to research. the public schools taught what was on the test.
the private schools looked at how I was doing individually and found ways to engage and challenge me. the public schools had a curriculum and if I didn't fit in it I got benched. case in point... entering 7th grade I read college level. in 7th grade reading class the work they gave us was killer dull and waaaay below my skill level, so I (stubborn thing that I am) wouldn't do the work. they addressed that by putting me (and my college level reading skills) in the slow-learner remedial reading class. and there I sat, not doing the even duller work in that class either. instead of trying to identify the problem (I was bored and underchallenged) they assumed I was unable. that sort of experience repeated itself more than once in public school, never in private school.
overall, the level of attention to individual students' needs was much lower in the public schools, although teacher-student ratios were essentially the same. don't know why, maybe it's the curriculum-focus of the public school, as opposed to the student-success-focus of the private school.
to my great good fortune, the public highschool I attended was flexible to a degree... I went to the counselors and was able to get them to let me take some classes independent study and on my own time so I could graduate a year early... but I was the driver behind that, not the school, not my parents (my mom says she didn't know I was graduating a year early until the graduation anouncement was sent to her 3 weeks before I graduated.)
private school taught me the skills to do that.