Too many families simply cannot afford to school their own children, the materials alone would be an overwhelming financial burden, many parents themselves lack education, and are in no way prepared to offer their children a useful learning environment, to do away with public education altogether would certainly result in severe class distinctions, and this is not something I think any of us want to see come about.
In my "perfect world" a community would build the infrastructure for a school system. Then a school board would take over to operate the buildings, fill the buildings with teachers, staff, students, and learning materials. At this point, parents who would like their kids educated would directly pay the school board - like we do with colleges.
When we purchase cars, we plan on paying for monthly car payments, fuel, and insurance.
When we purchase homes, we plan on paying monthly mortgages, home owners insurance, and property taxes.
Yet, when most people have kids - they just have 'em.
If people understood that they'd have to pay to literally pay for their kids schooling, and rearing, I'm sure we'd have a better society and less unwanted kids and a lot less social problems that we have today, ie. gangs