Hello Sergei,
Thanks for coming back with the solution. The power cycle fix (unplugging PSU for 10 minutes or holding power button 30 seconds) working on a Bluetooth issue is unusual but makes sense given what KB5077181 changed.
This update modified USB-related system files (USBHUB3.SYS and UsbPmApi.sys) which handle USB power management and hub control. Many Bluetooth adapters communicate through internal USB connections, so the power cycle essentially forced a complete hardware re-initialization that software driver reinstalls couldn't achieve.
I appreciate you posting what worked. This would be useful to others in the same situation.