In headsets like the Rift or Valve Index that send video streams over HDMI or DisplayPort that extra step isn't necessary; everything happens on the PC's GPU.
Ah, ok, that’s how I ended up with the original mistaken belief. I remembered “Timewarp is necessary” and “Timewarp can happen on the headset” and conflated them into “Timewarp necessarily has to happen on the headset”.
See Oculus Link over USB, for example: https://developer.oculus.com/blog/how-does-oculus-link-work-... The diagram there shows Timewarp running on both the PC and the headset side.
In headsets like the Rift or Valve Index that send video streams over HDMI or DisplayPort that extra step isn't necessary; everything happens on the PC's GPU.