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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.