OK, you establish all those checks, then run them on students, and 98% of them fail every single test. Now what? Can we change the past so that the background check can be fixed? Will we fit students with "better parents" assuming that anyone in this world would be crazy enough to adopt them?
In the past, in societies that were free of social engineering, such kids (and then people) were left alone, to mingle with others like them and live as they can. The remaining 2% did the same, but on their own level. But now a timid, smart kid has to coexist with a violent and possibly somewhat insane kid.
It's much easier to declare a classless society than actually create one. The experience of USSR tells us that it's just not possible, regardless of how much you pour into education and propaganda. Those things are largely defined by our genes, and very few people can overcome their natural instincts.
Then I guess 2% of them are stable enough to go to taxpayer-funded school. I don't see a problem with that.
Can we change the past so that the background check can be fixed? Will we fit students with "better parents" assuming that anyone in this world would be crazy enough to adopt them?
We accept that not every kid is a good candidate to be thrown into a minimally supervised, government run institution and let their parents sort it out. There are private schools, homeschooling, and information is more available than at anytime in the past. If a kid wants to learn, he will.