Formula One ( F1 )

but you give up a lot of control that you would otherwise have with a dedicated spring/damper system

What control would you be looking for? The job in this context is to keep the rubber on the road - and the tyre does this when the contact patch is massive and the pressure is comparatively low; it just deforms locally with little impact
on the overall pressure of the tyre because the volume is so large relative to the deformation. So I’m not sure what a rebound / compression system would add.

It’s not that the tyre is doing the suspension systems job, it’s doing a different job that you wouldn’t design the suspension system to do because you can’t really square the circle where it needs to deal with two opposing types of input whilst remaining light, predictable and reliable.

Would genuinely be interested of hearing any concrete cases where having the tyre doing more suspension work than the suspension system itself.

Go karts have no suspension systems at all :)