To be fair, this is a message from a member of the public rather than an official council statement, so maybe something got lost in the translation.
For example, press reports etc often use the term 'private' cars or drivers to refer to PHVs and PHDs.
But, of course, strictly speaking, when you refer to a 'private car' most of us mean a car that isn't an HC or PHV.
So maybe the council told this person that they had to use taxis or private hire cars, or similar, and the person conveyed that in the message above as simply a 'private car'. And maybe they don't know precisely what a private hire vehicle is - they may think it's what we would normally call a private car, or at least something that isn't plated.
(And which is maybe one good reason the press often refers to 'taxis' rather than private hire cars. I suspect some members of the public think a private hire car is in fact something that isn't plated, by dint of the word 'private', which of course technically refers to the method of hiring.)
And, particularly with regard to education departments and the like, often their terminology as regards stuff like this is a bit dodgy anyway. Ditto even licensing departments at times
So not sure if it can be automatically inferred from the message posted that KCC are condoning the use of unplated vehicles.