No one subsidises private cars. Can we stop funding private schools?, page-137

'well certainly funding of private education and health is a cause of higher taxation'

We'll just have to disagree on that one.
You aren't factoring in the impact of higher private costs forcing people into the public system that in turn creates a higher cost than the money saved.
People who paid fees to use the private system don't pay anything when they switch to the public system.
If the private schools received more government money than the public system you'd have a point but the opposite is true.