simple argument that a force applied farther from the rotation axis...

A.T.

I think you can simplify this to a King Truss (top left), just ignore the overhang at the sides:



Apply 2F downwards in the middle, and F upwards to the 2 sides nodes. It is a static case. Now your question translates to: Why doesn't it start rotating around one of the sides nodes? How does the one F on the other side balance the 2F in the middle?

Aside from the trivial symmetry argument, you can work out the internal forces here.

Aimless

Yes; that's about what I was trying to describe. This view of things is oversimplified, but at a qualitative level I think it captures the dynamics of how internal stresses on a microscopic level can translate into rotation on a macroscopic level.

From a teaching perspective, it might be worth it to construct a numerical simulation of a model along these lines (with tunable string constants) and allowing it to evolve in time. It shouldn't be all that difficult; certainly nothing compared to, say, a DFT calculation.