My understanding of this was that the OT clinic is commissioned by state schools to provide a service to them. They are not commissioned by private schools to provide that same service. Therefore, no service in this boy's case.
The state schools pay for the OT's services.
Of course a school who's not contributing to that contract, don't get the service.
It sounds to me the issue is the avenue in approaching the OT.
I imagine if the family approached the OT via GP they would of course receive the treatment needed.