Right, but you're describing a self-eating animal. If student success if the intention, diverting money from public schools won't help in the long run. It will work for a while, but as public schools get drained then private schools will be the only competitive option.

Then, of course, you would jack up the prices. Private schools are only limited in greed due to competition with the public sector. The more you erode the competition, the more expensive and lower quality private schools will become. Eventually we'll reach an inflection point, in which private schools are too expensive for vouchers, or our public funds would have to increase.

I also disagree on the institution not mattering. What many don't realize is there are virtually zero standards for private schools. Even today, many are not competitive. Rather, they exist as a way for insane parents to "educate" their children on fringe teachings. Sometimes that's religious schools, sometimes it's cult teachings, sometimes these private schools are more or less abuse centers or conversion camps.

SOME private schools have competitive education. It's not a given they have more competitive education, and I'd actually argue it's far less likely, because they have no rules anywhere saying what they have to do. Public schools are, at least, pretty reasonable in process and curriculum.

The reason conservatives are so keen on dismantling public education and pushing private schools isn't due to quality, although that's a convenient talking point. It's due to this lack of standards that allow conservative beliefs to flourish. It's often said education is an anti-conservative space. Naturally, the end goal is lower quality education, and this is simply the propaganda used to get to that desired end state.