>Is that what it is saying though? It is the difference that is being minimized, not the absolute difference. Seems to favor big gaps in favour of PE.

The difference is extremized (on average).
What I had in mind was Gauß principle of least contraint. That is if you can express your force as
$$F= - \nabla U$$ then you have some kind of balancing condition between a kinetic term and a potential term.