Seems to me that a frame that twists to the left in response to the pressure applied by the right foot, in the portion of the pedal stroke where power is being applied, must return more or less immediately to center during the "dead" part of the pedal stroke. Unclear how that lateral movement, back to the frame's original position, would transform into power being applied vertically through the left pedal during that pedal's power stroke.

Not saying it's impossible, just that it's hard to think of an obvious simple example of a lateral force contributing to the application of a vertical force.

Come to think of it, if the idea is that some kind of rebound could resupply previously applied force that would contribute to moving the chain, a more obvious move would be to use a shoe with a spongy or springy insole---perhaps a spongy insole (with a carefully controlled rebound rate) sandwiched between stiff inner and outer carbon soles.

(Maybe. Still struggling to grasp how and where the forces are being applied and retrieved during "planing.")

Dunno. Fashions in running shoes have gone from heavier maximalist designs to minimalist (and barefoot!) to super-shoes like the ultra-cushioned Nike Vaporfly in recent years. Will bike manufacturers similarly begin designing modern frames that incorporate elements promoting "planing"?

If planing exists, and genuinely contributes to pedaling efficiency (and thus would contribute to winning bike races), then engineers will eventually begin using computer-aided design to create frames with carbon layups that enable the frames to plane as well as or better than any steel frame ever built.

Unless those engineers discover that planing is intrinsic to the "soul" of steel bikes and, baffled, give up.