You could do all those things and you will still have people who can pay for better education for their children!
As long as people have choices there will be those who have the ability to buy better - be it homes, cars, holidays, health care or their children's education.
There will never be a level playing field. People aren't equal in all things to start with are they.
Some are richer or - better looking, healthy, intelligent, ambitious, diligent, committed, brave,
Etc etc etc
As I posted upthread even if you closed every private school what are you proposing to stop parents paying for private tuition after school?
It's no good wailing "Not fair" but having no sensible solution.
You sound rather Orwellian.