Dollars go to providers in each of those cases, but it's indirect. In K-12, the preeminent purpose of dollars is apparently to finance the (monopoly, public-sector) providers.
And then to Andrea's point, well...it doesn't work well-- and can't be expected to work well, with the current arrangements. Monopolies are known for high costs and taking advantage of customers. Govt is known for thick bureaucracies, wasteful spending, mediocre results, and inattention to the vulnerable. Let's fully implement the GI Bill for K-12 and watch great things happen.