> If it costs $1000/student/month irrespective of if a child goes to a public school or a private school, why does the state really care which one the child attends?

Because it doesn't cost a flat amount per student. The fact that funding is fixed per student doesn't mean that all students are equally costly to educate, and private schools that don't have a universal acceptance mandate impose selection criteria which tend to select for students which are less expensive to serve, increasing the per-student costs (but not funding!) in the public schools if public funds are used on an equal per-student basis to support students going to those private schools.