This is a fun thread. I'm in Europe, promotion of public school and restrictions on private schools, government help for housing, health care and a bunch of other benefits, we have that already
Anyways, I don't think reducing private schools is the solution. Up the level of all public schools instead. If public school could deliver high-quality teaching and the kids that got out of it could all do very well in college, then it won't matter if your kid comes out of a public school or a private one - he/she will do fine in college anyways.
Plus, aren't public schools a standard anyways ? If parents puts their kids in private school because private schools are much better than public schools, then everyone benefits from public school levels going up. (I also believe there's a maximum "level of efficiency" a point where you can't do better without overworking the kid, so if public schools' level go high enough, it will trivialize the difference between public schools and private schools too, and everyone will still be better off anyways.)
"Everyone should go to college" is not necessarily a good idea. I haven't lived it, but it seems that in the US, if you have a bachelor's or a master degree you're ahead of the game, and it actually helps getting a good job and a middle-class life. Here, many, many people have a licence (three years college degree) so it becomes trivial - you're just sitting on school benches for three years more. I don't really know how to get out of this problem - college degree and education is worth more when it's rare, so it can't benefit everyone. Anyways, developing trades is also a good way to live well, as it can pay surprisingly well.
Government help (health care, unemployment benefits, and any other social help) is very important as well. When you're poor, everything costs more in percentage of your income, so you have a lot less room to maneuver, and every setback is bigger and more crippling. If you have some sort of help from the government (or anywhere else really) you can put money aside and it makes going up easier since you don't have to spend all your energy and income on survival.