My understanding is that it uses Bluetooth just for pairing, so I'd assume that it's not so much an issue of "drivers" but more of controller firmware to get it to send inputs over Bluetooth at all.

Yes, the way it normally functions is directly connected to the Stadia server over wifi. It does show up as a controller connected over USB too. But Bluetooth is just for setting up the wifi connection.

I don't even want to call it pairing because the way you pair the controller to your game session is to enter in a 5 button combination that is displayed on your TV.

It works nicely as a controller over USB, I’m using my stadia controller with OpenEMU

I have empathy for their team on this topic. If they release bluetooth drivers they are committed to some minimal quality and compatibility standard and controller owners unaware of a larger OSS effort will be frustrated if it isn't actively maintained.

No they aren't.

Stadia shutting down means they're out of a job or moving internally to another project.

A driver being released from an abandoned project comes with no warranty

In the announcement Google said the tech will continue to be in operation for 3rd party partners - this suggests to me many aspects of the projects will continue even though it will no longer be available to public consumers.