There are good steps that can be taken, but objectively harming some students is not the way forward.
Improve public schools and try to get more money into the poor regions. Do not erase private ones.
Day school rates for good private schools (there are way, way more bad private schools than good ones) are in like the $20k-$60k/yr range. That's a lot of money to put toward avoiding public school, even if US private schools themselves are outlawed. You'd have to also mandate public school attendance, no alternatives whatsoever.
You'd also have to do something to prevent rich people from effectively buying whole school districts and turning them into private schools. There are already districts kinda like this—unleash the entire upper-middle and upper class on the current public school system, and pretty soon there will be a few dozen districts nationwide where it's impossible to buy a house for under a million dollars and the schools may as well be private schools.
As long as parents raise their own children, you're going to have crazy levels of inequality. If you also continue to respect personal autonomy and private property, you'll have ever crazier levels.
Cause rich families to blanket refuse to live anywhere near neighborhoods with poor kids and contribute massively to the segregation between well off and impoverished neighborhoods.
Just live in a place where every home in a 45 minute radius is over a million dollars to purchase.
This is a systemic problem, systemic measures will be taken, there will be winners and losers.
How does harming anyone help? No it does not depend on the some.
Life isn't a zero sum game. Pull people down and you don't distribute their value to others, you only destroy it.
"I am so, so tired of socialists who admit that the current system is a helltopian torturescape, then argue that we must prevent anyone from ever being able to escape it. Who promise that once the last alternative is closed off, once the last nice green place where a few people manage to hold off the miseries of the world is crushed, why then the helltopian torturescape will become a lovely utopia full of rainbows and unicorns. If you can make your system less miserable, make your system less miserable! Do it before forcing everyone else to participate in it under pain of imprisonment if they refuse! Forcing everyone to participate in your system and then making your system something other than a meat-grinder that takes in happy children and spits out dead-eyed traumatized eighteen-year-olds who have written 10,000 pages on symbolism in To Kill A Mockingbird and had zero normal happy experiences - is doing things super, super backwards!" ~Scott Alexander
Finland does that.