> My point is that public primary and secondary schooling seems to have been more effective (anecdotally) than private higher education.
Primary and secondary public education often fails to provide basic literacy competence. As a result, a high-school diploma isn't worth much, but the right college degree statistically is worth even the absurd prices that are now being paid for them.
> This is a strike against the idea that the government shouldn't be paying for it, in that in at least this one case the government paying for it seems to be working better than a non-government organization doing that doing so.
The government can pay for it without running it, see the Swedish model and the voucher system, which has resulted in improvements. However, teacher unions are obviously against this. They fear their jobs will be less secure if most parents had a real choice in where to enroll their children.