Your last paragraph is exactly what the DC superintendent proposed. The teachers' unions were so threatened that they didn't even allow it to come to a vote.
Of course, things like that are a lot harder to mandate from the federal level. You have to get into things like measurement and (opinion here) most of the really important things about teaching are very hard to measure.
Maybe you could have a variable-value voucher system where the amount a school gets for having taught a kid depends on test scores at the end.
I'm all about charter schools in theory, and still am much of the time in practice, but often the way the funding is structured is ridiculous.
Back when I was in local politics, a charter school opened up in our town. For every kid who enrolled, we had to pay the charter school more than we were paying to educate the kid ourselves. AND bus them there, presumably to add insult to injury. So I'm a little suspicious of vouchers and charter schools even though they instinctively sound like good ideas to me.
Presumably you'd be making some assessment of value added, which would be hard of scores alone. A school in an area of high illiteracy that manages to improve literacy levels of pupils could arguably have added more value than a school in a high achievement area where achievement levels are pretty constant - moving people from a fail in Maths/English to a pass might impact their lives far more than moving them from a B to an A (or A to A* as they're now called in the UK; grade inflation FTW).
Could lead to increased stratification. On the other hand, is it worth spending a lot of money to educate a student who is dumb as a doornail anwyay?