From my side I can add, that I also use ICE plus I have my PC connected to TV via HDMI.
I have a SBP setup, where Big Picture target is TV plus I use autohotkey script to switch audio source automatically:
https://github.com/barat/autohotkey-scripts/blob/master/steamBigPictureAudioDeviceSwitch.ahk
Thanks to this, if my PC is turned ON, I just switch TV source to HDMI2, grab Xbox One controller, press "X" button (this on the middle of the gamepad) twice and SBP starts on TV screen with audio source switched to TV as well. When I close SBP, audio is returning back to PC speakers, and Windows main screen is switched back to monitor as well
Almost like console.
I have a SBP setup, where Big Picture target is TV plus I use autohotkey script to switch audio source automatically:
https://github.com/barat/autohotkey-scripts/blob/master/steamBigPictureAudioDeviceSwitch.ahk
Thanks to this, if my PC is turned ON, I just switch TV source to HDMI2, grab Xbox One controller, press "X" button (this on the middle of the gamepad) twice and SBP starts on TV screen with audio source switched to TV as well. When I close SBP, audio is returning back to PC speakers, and Windows main screen is switched back to monitor as well

Almost like console.
Windows 11 Pro| Gigabyte x570 Aorus pro | 5800x3d | 2x16GB 3600cl14 |Gigabyte RTX 2080 8GB Gaming OC | Wii Bluetooth Module (WML-C43)
2x Wii Remote RVL-CNT-01 with MotionPlus | 2x Xbox One controller
2x Wii Remote RVL-CNT-01 with MotionPlus | 2x Xbox One controller
