Yes I skipped 90% of this. But I do see it may be past time for serious answers and maybe this has been answered as well.
If a whole system is elastic springs, then physics says (claims?) energy has to be conserved. But the rider is not an elastic spring. If the rider puts out effort to deflect the bike, the rider may not get that energy back when the bike relaxes. It's hard to analyze a full bike, so I think it might make sense to try to imagine the simplest system possible.
- Assume a perfectly efficient, perfectly stiff bicycle, except for the handle bars. Start with the bike coasting, rider standing, equal weight on the pedals, zero weight on the handlebars.
- Assume the rider keeps their body stationary put reduces weight on one pedal and increases on the other. This provides torque at the crank, but also tries to lean the bike. The rider then pulls up on one bar and pushes down on the other, so the bike and the riders torso are still stationary, but the bike stays balanced and is propelled forward.
- The forces on the bar will flex one side up and the other down, but elastically - no energy lost. Just a conversion from work energy to potential.
- When the pedals turn the rider will reverse the forces and the bar will flex the other way. The bar gives up it's potential energy in one direction and stores the same amount in the other direction.
- However, when the bike feeds energy back into the rider arms, the arms can't store that energy. Your lungs understand somewhat the energy difference between force as you contract a muscle vs extending it. That's why your cardio system works so much harder hiking uphill vs down. One direction burns sugar and oxygen and the other doesn't, but feeding energy into a muscle doesn't unburn the sugar.
- Conclusion: Flex may not make the bike less efficient, but it can make the rider less efficient. The flexible bike may conserve energy, but the resulting additional motion of the rider can waste it. At least in this simplified case. At a minimum we can say that at least some types of bike flex will waste energy. Stiffness matters.