All right. Here is my criticism of the concept. In order to store energy, movement is required. A bike frame - even a noodly one - is stiff. Now that we have a bowyer in the conversation, imagine a bike frame strung like a bow. How far could you pull it back? How far would the arrow go? Not far, because the bike frame cannot store very much energy, because it's stiff. A spring that barely moves is storing barely any energy. When you pedal a bike you are putting in a force similar in magnitude to pulling a bowstring. This is enough to sometimes bend some bikes enough the wheel rubs the brake pad. That's like a millimeter, or a fraction of a degree. A bow, which is actually flexible and does return a lot of energy, lets the arrow slide about two feet. A spring moves in proportion to the force and stores energy in proportion to the movement squared. A spring that is 2x stiffer stores a quarter of the energy for the same force. A bike frame that is 100x stiffer than the bow stores 0.001% of the energy. It's just not enough to change the resistance profile enough to account for a 4% much less 12%.

Here's an experiment many roadies could pull off. Put the front wheel in a trainer and nose it up against a wall. Put the pedal at the first click over 3 o'clock. Prop a ruler up on the back side of the pedal and set up your phone to film the ruler. Stand on the pedal. Your full weight is as much torque as you will ever give it. Each 1/8th inch of movement of the crank is one degree that it could be "giving back"