While there is "free choice" in picking between public or private institutions, that's where the "free" stops.
Full disclosure @Mike3094... are you a public school teacher or related to the public school system in some way?
I'd suggest that "... for the government to pick up the tab." is hardly the issue. It's have the government not take money out of our pockets if we don't use a service they provide. It's redistribution, in another form.