How do you not run a public school that is charged with serving all students in its district as a social program?

You run it as a school, since that's what a school is for. Otherwise you fail at both being a school and a poor replacement for the social programs you're trying to do on the side.

For example - a school is not a correction facility. If a student is violent, they do not belong in school. Get them out and put into a proper facility designed for such things. It's not the school's problem anymore to think about.

The results speak for themselves on the topic.

A school is to educate students who are there, present, and want to be educated. They will fail at any other task.