I disagree with that prediction. I think that, if the other choice were having their own children at home with them all the time, many parents would make the decision to send them to school, either paying for it themselves (as they do for private daycare now) or applying for charity.
The charity might well come with provisions that the parents improve themselves, or at least support their children's education, with positive results for two generations.
If for some reasons all schools disappeared in a blink, I would hope you would be right and I would be wrong.
But I have no faith in a huge portion of our society to do the right thing, it’s nice that you do though. I’m just a cynic I guess.