That's true, I feel stupid for not having thought of that!
I'm not an electrical engineer, but with the FETs that modern Intel chips are using, what fraction of their power consumption comes from parasitic gate capacitance, versus other losses?
And if you operated in the linear region, what's the ballpark steady-state I_SD current you'd need on one FET to drive the gate of the next FET?
I think that's what this comes down to: if gate capacitance dominates other losses even in the linear regime, you still win by not having a clock and lots of digital transitions.
You could even imagine exploiting that: apply 'slow' augmentations of the input data that get you the equivalent of a bunch of iterations on a single example batch, while incurring a much smaller fraction of that initial cost because activations aren't going to change nearly as much as switching to a whole new example batch.