Centralization and size comes with inefficiency and slow decision making. It also comes with economies of scale. For something like the Medicare, the economy of scale dominates and it makes much more sense to be federal. When it comes to running schools, there really isn't much economy of scale so it makes much more sense to run it locally.
Economies (and diseconomies) of scale exist, of course, but aren't central to what is the responsibility of the federal government and what is the responsibility of the state governments. That, of course, is laid out rather plainly in the founding legal document of the country.
