It's not as apple & orange as you think. Schools have to teach the same general curriculum whether they are public, private, or parochial. In 11th grade, you learn U.S. History no matter where you are.
As far as parental involvement, I'll allow that there may be more with private schools. However, my mother-in-law has taught Catholic grade school for over 30 years and there are plenty of parents she deals with who don't know and/or don't care how their kids are doing in school. In fact, some parents feel like they don't have to get more involved precisely because they are paying so much for a private school. On the other hand, my father-in-law has been a career public school teacher and he says the same thing; some parents really care and some don't. Parental involvement and socioeconomic status is a factor, but not enough to hang your hat on.
As far as screening students, that doesn't really happen in grade school. My Catholic grade school took all comers, there wasn't an entrance exam. And the only way most private schools "get rid" of underachievers is by those kids flunking out, which also happens at public school. I can't recall a single kid getting booted out of my grade school for not performing highly enough. You can scrape by in private school just as in public. High schools can cherry pick, but that some students don't make the cut says more about the quality of education they received prior to trying to get in. A good education gives you a leg up no matter where you get it; but it's likely to be better if it was from a private school
Overall, private schools are just doing it better than their public counterparts and I think it's something systematic that goes beyond money or parental involvement or cherry-picking good students.
All that being said, I do not support school vouchers or school choice. My feeling is, fix the damn schools! I would like to see a good public school system that performs closely with private schools. I'd hate to be in the position that my parents were. They sent me to Catholic grade school, but the parish high school closed and there wasn't another Catholic high school for 50 miles. So, they had to send my brother and I to public high school and hope for the best. I don't want to eliminate public education, I want it to be better so that this debate doesn't even exist anymore.