It is hard to define an "intuitive explanation" for why something is true, but one knows it when it is at hand.
Example: Why does a pin puncture a strong rubber sheet? The force is applied to a very small area, so the pressure $F/A$ is enormous.
Of course, intuition is elastic, it can be trained, and such explanations might or might not exist.
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Now suppose that someone came up with a theory for rotational motion which postulated that the square root of torque divided by the cube of the moment of inertia is minimized along all paths. Let call this magnitude $Z$. Now I do have a physical intuition for what torque is, and what the moment of inertia is, but it's hard to get a _physical_ sense of what $Z$ could be designating -- it just looks like a mathematical combination. On the other hand, $(1/2) m v^2$ is also a mathematical combination, but I have a physical sense of what it quantifies.
That was my question about the meaning of action $T - V$. It feels like a mathematical combination, not a physical concept. So, although the principle of least action is elegant and powerful, I don't find the statement of it to be physically intuitive.
I could say that, one-hundred times over, for quantum theory.
That may be just the way the cookie crumbles. And it may say more about the subjectivity of intuition -- which is conditioned by evolution and experience -- than about the physics itself.