The effects you list are accurate, but they don't really enforce your main point. Just because public schools do mobilize kids out of their bad homes (by making it a parent's legal responsibility) doesn't mean it exists for this purpose.

Schools are a place for kids to learn. It's not a place to get a meal, nor a solution to a bad home life. We ensure we provide these outcomes in schools today but that's only because society/govt has failed to get these things accomplished in a more appropriate manner.