It seems evident to me that tendons have evolved to be elastic to help prevent rupture during that sudden application of high strain – in other words, when the owner lands on the feet after falling or jumping. Elasticity spreads the strain placed on the tendon and its muscle over a longer period of time, avoiding possible injury from the sudden high stress/strain of impact.
It is important to note that no spring or elastic material returns all of the force applied to it in recoil, so, clearly, a stretched tendon does not recoil with more energy than that required to stretch it.
Note that shock absorbers in bikes and cars are there for comfort and to prevent damage to parts and drivers. At stiff-framed bike is more efficient at transmitting the rider’s force into forward movement; a flexible frame loses energy, and is slower to accelerate — so the human body with its tendons and muscles.
We can see that any rapid strain on the tendon will cause it to stretch. In downhill running, we’d expect to see this stretch on landing. On the flats, we’d only see this stretch at high speed and during rapid acceleration, such as in sprinting. Starting from the blocks in sprinting, for example, is a time when rapid, extreme strain is placed on the Achilles tendon. The tendon will recover when the strain on it is reduced, at the end of extension. The tendon will not recoil with the energy that took to compress it, but it may be expected to give something back at the end of extension. However, more force could be delivered during extension if the tendon where rigid. The injury risk would be higher, though.
Which brings us to shoes and experiments to create “spring”. One hopes that cushioned shoes are made from dense rubber that can return part of what it gives away in hard landing or in hard extension. But no shoe is going to make you faster by giving you more than you gave.