Will Bluetooth ever be enabled on the Stadia controller?
"Ever" is a long time. The hardware appears to be there, but is currently only used for setup (much like the Bluetooth hardware built into the Chromecast). The question is, what use-case are you interested in supporting? Are you wanting to use the Stadia controller over Bluetooth to a PC or phone? As of today, you can use the controller wirelessly over WiFi instead. Or are you wanting to connect Bluetooth headphones to the controller for audio? That's something that would make sense, I agree.
Headphones for sure!
I found a way to do bluetooth headphones on chromecast ultra. I made a video on YouTube https://youtu.be/xwg3brh7_bs