Woven metal tires are old. And nitinol is also old (it was not developed for the rover, a wiki check would show as much).

But the combination is new. I imagine the conversation went 'hey I am working on chainmail tires' 'areny those heavy' 'well. We need them light, strong and elastic, and you only get two...' ' hol up, there is an option, you heard of nitinol?'

And thus, cool history.

They developed the shape memory part recently while developing a new wheel for the Mars rover.

From your NASA link:

“ In one particular moment of serendipity, Engineer Colin Creager and Materials Scientist Santo Padula had a conversation that completely changed the path forward.

The game changing material that dramatically advanced the development of spring tires was nickel titanium, a shape memory alloy with amazing capabilities as explained by Santo Padula.”

This video has a clear explanation of the shape metal alloy on the Mars rover wheel: https://youtu.be/2lv6Vs12jLc

Short answer is that the material is super elastic, and will return to its original shape either when heat is applied, or when stress is applied.

Nickel-titanium ("nitinol") shape-memory alloys are also "superelastic", which means they have an additional range of deformation past normal elastic deformation before plastic deformation from which they return back to their original shape. So, not just spring steel.

Not a shape alloy expert, but the temperature that causes them to morph can be manipulated when they're formed (forged?). I have a few strands that just the heat from your hand is often enough to trigger them.

So, the heat from friction is potentially an option.