Here are some more details. My laptop is ASUS ROG GL502VS. I tried this with Jabra Elite 65t and Jabra Stealth, both of which support A2DP. The pairing goes smoothly and everything seems to be installed correctly.
Note that the stereo device is present as well as the Audio Sink service in the headset. However the Headphone audio output is "disconnected".
Right-clicking on the Headphones entry and choosing "Connect" does absolutely nothing. No error messages, no reaction at all. The entry stays disconnected. Disconnecting the Handsfree first does not help. Unticking the Handsfree Telephony service in the device properties does not help. Rebooting, re-pairing, installing the latest driver, clean reinstall of the driver via safe mode does not help.
A bit of history
This worked fine a few days ago. Then at some point I turned on my headphones and got a Blue Screen of Death. I blame some automatic update because I didn't update anything myself. I only uninstalled Razer Synapse before that happened, which might or might not be a coincidence. I couldn't connect to Bluetooth Stereo after that.