I have had this exact issue for the past month and i didn't know why (because I used my BT headphones fine for MONTHS.)
MY FIX (If you have intel and use/update with their drivers. Specifically me updating using Intel Driver & Support Assistant)
- Go to Device Manager
- Bluetooth
- DELETE Intel Smart Sound Technology
(I kept the Intel Smart Sound Technology under the actual sound settings because i don't trust it not breaking)
- Restart your PC
This should allow Windows 11 to auto switch to hands free without issue. I finally fixed it after a month of trying to figure out why this was happening. Hope this helps!