I think there is the underlying notion that among any set of possible paths, it is always possible to find for each path a quantity for which a particular path

I think there is the underlying notion that among any set of possible paths, it is always possible to find for each path a quantity for which a particular path is optimal. That seems a bit trivial to state, but reasoning on the space of those quantities is more flexible. It doesn't mind the non-linearity between parameters and resulting paths as much. So when you are trying to find the laws of physics, reasoning on that higher level space of quantities-that-will-get-minimized seems to simplify things.