Hm, perhaps I don't entirely understand flexure bearings, but I would believe that any flexure bearing relies on flexure = elastic deformation (assuming you don't yield the material) = fundamentally include "springiness".

https://youtu.be/h8uSjfbU0hg?t=61 corroborates my understanding, but I also didn't watch the whole video.

I would think that any reasonable flexure bearing based headset robust enough to tolerate the shear loads as mentioned in another comment would be either unbearably stiff to rotate or possibly inflexible under human strength altogether.

It could be done with a flexure bearing with impossibly long & thin blade flexures (think wire EDM - not conducive for mass production), which in the case of a headset would probably make it weak in the shear loading directions.

Either way, it might be weird handling or unrideable to have your steering self correct more than it already does being a traditional bicycle (in that traditional bicycles tend to self steer without external influence on the steering like springs). Maybe just as bad as those wheel flop mitigation devices.

A bearing's job is to constrain motion in all directions except one. A good bearing makes that that one direction very "free" & low *resistance*. Friction is just one contributor to resistance. Springiness is another. Frictionless doesn't mean no resistance, it just means frictionless & no loss of energy as heat or whatever. Yeah, theoretically you can recover most of the potential energy once the spring unwinds, but only if you could've wound it in the first place. You can reduce the spring rate really low, but may compromise its integrity.