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I don't know the specific thought experiment, but generally, matter tries to follow spacetime geodesics if nothing forces it not to. So if a gravitational wave passes though a gas or other medium in which the particles can move freely, I would expect the gravitational waves not to produce work. I'd expect that to be the case in the early universe (though I'm definitely no expert in that).

But beads and strings are definitively not in that category; rather they are solid matter where the relative locations are mostly determined by the electromagnetic forces acting between them, and the corresponding quantum properties.

I think a good analogy should be tidal friction. Tidal friction happens when a massive, solid object rotates in an inhomogeneous gravitational field, that is, its solidity forces its atoms to move along trajectories that do not agree with the spacetime geodesics they would otherwise follow. The tidal forces cause changing deformations as the body rotates (and relative movements, e.g. of the water relative to the earth), and since those deformations are not perfectly elastic, that means friction.