OK, I tried all this, didn't have any success, then was consumed by Christmas. However, I have now solved the problem!
I have a desktop machine with a Bluetooth USB dongle in it. Critically, this was installed with the OEM drivers, which work in every other respect for devices using the usual BT services for hands-free audio. I started having some problems with automatically connecting to one set of headphones, did some research and realised on the way that the way W10 presents BT connectivity through my laptop machines (BT built-in) - ie in the notification bar buttons - was absent on this machine, which showed the 'old style' BT icon in the system tray.
The Xbox app - and I suspect other W10 apps with audio - will not recognise BT Services for audio unless installed with W10 drivers that surface the capability in the notification bar it appears - which appears to require some degree of fiddling for W10 to 'correctly' recognise that it is BT enabled - this appears to be a common complaint from people using USB BT dongles.
What I did (copied from elsewhere):
1. Turn on Bluetooth Windows Services
Open windows services (run services.msc or search for "services" on windows search).
Change the Bluetooth Handsfree Service and the Bluetooth Support Service to automatic (on properties) and start them.
2. Uninstall the bluetooth driver that comes with the device
When you connect the device, the driver gets installed automatically.
Go to Device Manager (Right click on Start Menu and choose Device Manager or Search for Device Manager).
Find your device on the list. My device was on Universal Serial Bus Devices (I think), named CSR8510 A10.
Right click, choose Uninstall and remember to choose "Delete the driver software for this device" (I tried without this and it did not work).
This surfaced the notification bar button - and solved the problem playing XBox games.