[0] Obviously the problem of what %age to give to each camp is an important second-order problem, but if you imagine a scenario like a university, I find it more likely the Provost's office can figure out how much to give to which Dean than they to be able to figure out how much to give to which lab.
(I also think you're right to observe that even within a particular camp of a particular field there will be relationship-driven gladhanding that helps determine the flow of money, and that's a real concern, but it's just a more localized concern than I think you give it credit for. I'm also not entirely sold on it.)