I agree that all of your explanations fall well on my metaphorical ears. They seem right, and I can nod along while I read them.
But I can't help but have a nagging suspicion that you'd get extremely similar results if you just tossed a random sample of kids into a school after filtering for household income > $XXX,XXX/yr.
I'd also argue that even if you compared gender segregated schools vs a mixed-gender private school of the same price, you'd still be left with a confounding variable of "whose parents clearly valued education enough that they either specifically picked, or tolerated (depending on their views on this matter) a gender segregated school.
Remember that parental involvement is a very powerful predictor of academic success (though I'll admit I don't know if it trumps the predictive power of just socioeconomic class).