I would expect any random headset to plug into the headset and microphone ports and Just Work, and ditto for USB

For the most part these days, they do. But OP asked about wireless.

or Bluetooth headsets that report themselves as the appropriate device class.

The problem with Bluetooth is not the operating system or drivers, but Bluetooth itself. The spec famously lacks provisions for good quality stereo output with good quality input at the same time. This is why many wireless headsets use a (non-Bluetooth) dongle.