Lots of public goods improve when competition is allowed to happen. Look at freight trains, which experienced a boom in profit, after the government stepped back and allowed them to operate in a free market (circa 1970 onward). Ditto phones when the ATT monopoly was broken up (1980 onward). And modems - after 30 years of stagnation at a slow 1 kbit/s, they rapidly grew from 1k to 56k once the ATT monopoly was lifted.
Postal service improved dramatically when the Laws barring UPS and FedEx from carrying goods were lifted. And then there's the military..... and yes I know you hate the military..... but it too has competition. There are dozens of contractors fighting against one another to win the right to build the next government plane, or engine, or hydraulic pump.
There's no reason we can't do the same with schools, where Microsoft High versus Apple High versus Penn State High (et cetera) compete to attract students and win those valuable government dollars. - Just like what happens in some European countries (the money follows the student to wherever he or she goes, even if it's a private or catholic school).
Basically the K-12 environment should be run the same way the College system is run. Students choose anywhere they want to attend, and colleges compete with one another, which raises the bar to a high level. IMHO.