TL;DR
Make sure your Intel Wireless Bluetooth driver is at least of version 20.90, and enable all Bluetooth services for your headset device.
Enable the needed Bluetooth Services on the Device
The device doesn't appear as an audio device in spite of being connected, because its audio-related services are not enabled. Navigate in Explorer to Control Panel\Hardware and Sound\Devices and Printers, right-click on the headset device, and select "Properties":
In the device properties, the Bluetooth services related to audio are likely to be unchecked:
Check all the services, and click "OK".
There are two possible outcomes now:
The services will be enabled and the device will be recognized as a sound device.
"The Supplied User Buffer is Invalid" error dialog box will appear, once per each device.
Fixing the Supplied User Buffer is Invalid error: Bluetooth Radio (host) Driver Update
If you get the error, the typical cause is an outdated Bluetooth radio hardware driver. As of 12/2018, the Intel driver version 19 is seemingly too old to reliably operate with the newest Windows Bluetooth stack.
The newest drivers for Intel(R) Wireless Bluetooth(R) must be downloaded and installed from [Intel's site]. I chose the newest driver at this time: version 20.90.
After installing the driver, power cycle the headset, ensure that it shows connected in the notification center's "Connect" pane, and re-enable the services as described above. After clicking "OK" to re-enable the services, the headset will become available for use as an audio output.
