I'm not sure that's true. In this area, the best private schools charge more than college tuition, and more than the average per-pupil expenditure of the public schools, but their teachers are paid less and have fewer benefits than public school teachers.
You're assuming that all the money that goes to public schools now would be put into vouchers, I think, which doesn't seem to be the way it works in areas where vouchers are currently available.
I'm guessing that people who are now willing and able to pay private school tuition would still be willing to supplement the voucher cost (which might keep their private schools more exclusive), but people who can't afford private schools now might still be relegated to sub-par schools.
That's all speculation, of course.
The vouchers would really help all those lower-class parents who are determined that their kids get ahead. They are a minority, but a significant one.