Why does the 'sticky bead argument' for (gravitational waves carrying energy) work?

I would guess that the argument applies to the case where the frequency of the gravitational wave is low compared to the natural frequency of the stick, or put another way, the rate of change of length due to the gravitational wave is small compared to the speed of sound in the stick.

If you take some line normal to the GW and of length $\ell_0$ then the GW will produce an oscillating strain something like:

$$ \gamma(t) = \gamma_0 \sin (\omega t) $$

and the length of the line will be:

$$ \ell(t) = \ell_0 \left(1 + \gamma \right) = \ell_0 \left(1 + \gamma_0 \sin (\omega t) \right) $$

So the rate of change of the length of the line will be:

$$ \frac{d\ell}{dt} = \ell_0 \omega \gamma_0 \cos (\omega t) $$

In other words, a point at one end of the line will move relative to a point at the other end of the line with a relative velocity $v_r = \ell_0 \omega \gamma_0 \cos (\omega t)$.

Now lay down our stick along the line $\ell$. If $v_r$ is much greater than the speed of sound in the stick the intermolecular forces acting in the stick cannot act fast enough to stop the stick stretching and shrinking with the gravitational wave. That means the stick and bead will move together and the bead won't slide.

Conversely, if $v_r$ is much less than the speed of sound in the stick then the intermolecular forces will resist the gravitational wave and stay the same length. In that case the stick will move relative to the bead and we'll see the relative motion and associated frictional losses that Feynmann describes.

Since the frequency and intensity of the gravitational wave are probably outside our control the only variable we can change is the length of the stick. If we make the stick long enough it will oscillate with the GW and if we make it short enough it will stay the same length.

One way to think about it in more intuitive terms than comoving/proper is to drop down to the micro level and consider how molecules hold their shape in the first place. They are bound together by stable electron configurations; which in turn obtain their shapes from quantum-electro-dynamical processes; an interfering sum of photons bouncing back and forth between the charged components of the system.

So the resting bond distance between (wlog) two hydrogen atoms is the result of some interference pattern between virtual photons moving back and forth around those protons and electrons. The propagation of those photons from one location to the next is metric-dependent. One can think of a gravitational wave passing as locally slowing down or speeding up light; or 'inserting' or 'removing' units of length locally. So such a modification of the metric should shift the interference pattern of virtual photons that keeps the molecule together; in such a way as to push the atoms back in their 'neutral' position; which is the position that keeps the integral of the metric between the atoms at whatever fixed value characteristic of H2.

Another way of putting it: if the metric expands, a photon going from atom one to atom two will arrive 'late', with a bigger phase shift, and the net effect of this is really indistinguishable from stretching the atomic bond by more conventional means.

So indeed if the wave is slow relative to the natural frequency of two masses constrained by a rod, the masses will move relative to two masses not so constrained; and if the wave is fast relative to the natural frequency, a plain old strain gauge should register a signal.