What a screwed up system!
All schools here are attached to a parish or, if a regional school, to the parishes in the region. The Pastors are responsible for the operation of the schools. Both the parishes and diocese provide support, with the diocese focusing support on poor inner city schools that are not self-supporting.
Every parish either has a school or is affiliated with one, so that everyone knows where their children are supposed to be sent.
I do wish the system was different. It is very difficult to come up with $25,000/year for my children to attend Catholic School. It is the best money I have ever spent and would sacrifice much before removing them. I am not sure why it is this way, but I am sure that there is a reason. I would prefer the Philadelphia system and $4500/year.