To say that flex involves energy loss is to say that some of the energy applied by your legs is converted to heat, from an engineering or physics point of view. For this to happen requires that the objects flexing are what engineers call "inelastic." Inelastic materials flex by converting energy to heat, and that energy is truly lost to the mechanical system. Conservation of energy still applies, that lost energy would be converted to heat. But the steel of bicycle frames, and all steels, is highly elastic like a spring. The spring, and the bike frame is springy, bends by storing energy in the shape change of the steel tube (it's small, but it's there), and returns it when the tube springs back. It springs back when the next half pedal stroke occurs. This is all because it IS a steel frame and steel does not significantly convert energy due to flexing into heat - no steel does this. It doesn't lose energy, it stores energy and returns it on a cyclic basis.

I think what people have been seeing is that if you press down hard on the pedal on a flexy bike, less power is transmitted to the chain at the moment you hit it hard. So a less flexy bike will have snappier pedal response. But if it doesn't translate to energy dissipation over time, speed is not reduced over a distance, since speed is a result of the rate of energy transfer, also known as power transfer. At least this is a simple physics interpretation. It's not without problems, however.

Now that I've opened the door, I'm gonna duck and watch what happens.