Doesn't work in practice, though - you end up with schools "teaching to the tests", to the detriment of other unscored qualities.

The obvious solution: more tests, and more bureaucracy. By the time you've got enough tests you might as well have a state-run school.

What do we need the market for then?

To minimize cost. A better example than education would be carbon emissions. Suppose we want to reduce our carbon emission to X tons per year. If we auction off X credits permitting 1 ton of carbon each, the market will find the cheapest way to reduce our emissions to X. Simply being able to measure emissions would allow us to reduce emissions (say, by requiring all companies to reduce by Y%), however, this is almost defiantly going to be more expansive.