I think one big problem is that low test scores are highly correlated to poverty. Parents who earn less usually don't have time to help kids with their school work, or don't understand the school work, or don't know how to study or teach children. Sure, there's exceptions to this everywhere, but that's the general pattern.
Incidentally, Louisiana has/had a program called TOPs that covers in-state tuition for students that get over a 3.something GPA. Who benefits the most from it? Kids whose families make above the median income in the state.
I don't think giving X dollars to the families with the top ranking students would change society overnight.