Hmmm .... Or, when the frame flexes, some of the energy which would have gone to turning the wheel, is lost in the system.

"Back into the system" ... What system? The human power system? The drivetrain? it is sort of like how old muscle cars used to have "axle-tramp" where the engines produced enough torque to wind up the springs and wrap the springs around the axles---so track bars or Lake bars were used to stop the flexing.

I am not a scientist; if you don't like what I say, ignore it. But in every discipline I have ever heard about, flex absorbs energy, dissipates energy ---it is not "returned" anywhere. The energy is used to move something which shouldn't be moving (or doesn't help the vehicle move) and is converted eventually to heat and friction and stretching and bending ... the energy is absorbed by the motion, and is wasted in terms of propulsion.

It takes energy to flex a bike frame--energy which is not moving the bike forward, because the flexion of the frame in no way transfers rotational energy to the rear hub.

Pretty simple example ... would you want to hammer nails all day with a hammer that flexed? When you saw the guy next to you who had a real hammer, driving each nail with X strokes, while you needed X*1.5 blows because your hammer flexed ... any talk about "energy returning to the system" wouldn't make you any less tired. You'd go out and get yourself a hammer which actually transmitted power to the nail.