Originally Posted by Kapusta

The flaw in that argument as I see it is that even if the frame does act as a perfect spring, and returns all the energy stored in it when flexed, it does not necessarily mean it returns the energy in the most useful way. For example, since the frame is releasing the energy at the same time as you are easing up on the force on the pedals, it may go into slowing your pedal stroke down more than making the bike go faster. In other words, the energy may get transferred from a very efficient spring (the frame) to a terribly inefficient spring (your legs and body).