First, the goal
the SOLE PURPOSE of any educating entity (be it teachers themselves, or the very building they work inside) should be to educate. THAT IS ALL.
Other priorities creates inefficiency to that main goal
The idea of a "ruling class" is millennia old. Every parent wants to give their child every advantage possible, but in the end that leaves some children lacking.
The elimination of private education would in a sense throw every child into the same kettle...And those parents with influence will be forced to take a closer look at what public schools are lacking and use that influence to change and improve them.
If you want to open a private school, the government should not be able to stop you. It is really as simple as that. I dont see what benefit banning private schools would have. If you want to send your kid to a private school, send them to a private school
Schools are meant to nurture creativity and new ideas among students. Under different board members, each school is run differently and there are variations seen in the curriculum. Each academic council prescribes a syllabus for each class, which it thinks best. Now the opinion of everyone is not the same and thereby what is good for one may not be, for another.
If all schools are run by the state, there would be an uniform code with respect to curriculum and maybe, even with co-curricular activities. That would kill the budding genius in each child for we do not know in which field the child excels in.