In Engineering we accept that a spring when used within its elastic range is 100% efficient. Elastic range means we do not disturb the atomic bonds and there is no permanent deformation. The force to compress a spring is linear to its change in length, meaning all the work as energy going into the spring is stored. It follows that when the force is removed from the spring, it returns all the energy stored as it decompresses.

A frame that deflects due to lack of stiffness, essentially behaves exactly the same as a spring.

In practice, if a cyclist is exerting maximum force on the pedal at say 3 o'clock, the frame will absorb some of that work (energy) as it flexes, it will be lost to going through the chain to the back wheel. As the pedal stroke progresses though, at say somewhere between 5 and 9' O clock, the frame will lose its deflection and return to it's normal unstressed position. When that happens, it returns the work, or energy into the drive train.

Another way to look at this is that even the stiffest frame on the market flexes, even if that flex is only a thousandth of an inch. It means all we have done, is increased the spring rate to a much stiffer spring, but it will still absorb and release the same energy.

The fact that a spring is 100% efficient means there is no energy loss.

There are practical limits though, on one of my old classic noodles, the back wheel loses traction from side to side under a full power sprint, which means energy is lost.

Riding with a flat tire as posted above is different, as we are increasing rolling resistance exponentially and turning it into heat absorbed by the tire. Big energy loss.

In short, fatter tubes save weight by producing the same strength with less material. I can remember way back never getting more than two seasons on a steel frame, before it cracks, somewhere. Do some research and see some folks with their light weight Trek's complaining about cracks. It is the strength that is important.