There are a limited number of spots in private schools, if everyone suddenly has $x more dollars to spend on private schools the private schools will raise prices by at least $x to reduce demand. The wealthy will still be the ones in private schools and tax payer money is diverted to the private schools instead of the public schools while the poorest are still in public schools.

There are also other hurdles to poor kids attending private schools, for example private schools don't have school busses, so they are reliant on kids having parents with the ability to drop their kids off and pick them up every day.