At the moment, the balance is still very uneven. The masses send their kids to public school. The more attentive, involved, educated parents send their kids to private school, and a few of the less fortunate find ways to get their kids in too, and those few kids benefit BECAUSE they are few.
If the situation changes such that all parents can use vouchers to switch over to private schools, the private schools will have essentially BECOME public schools, and as such, the quality of students will... change (I'll put that delicately.)
Now, any private school that resists the lure of easy money and upholds the standards for enrolling and expelling will be okay... it's just that from what I have seen here in LA, they can't resist. Suddenly, all this money comes rolling in if you accept these "other" students.
And here's an extra grim warning: women with problem boys (usually single moms) that have been kicked out of several schools are particularly drawn to private schools. Just letting you know.
If the religious schools have a crack at vouchers that could change your scenario I would think. And magnent schools, as long as there is a core Western Civ requirement would also.