The problem with the game controller connected via Bluetooth (Win11)

I know its late but it will definitely help someone else. 
This seems to be a known Bluetooth issue in Windows 11. A lot of users with Xbox One controllers have reported the same thing,  once Bluetooth is turned off and back on, the controller won’t reconnect until the PC is restarted.

You can try removing the controller from Bluetooth devices and pairing it again after re-enabling Bluetooth. It’s also worth checking in Device Manager under your Bluetooth adapter settings and disabling the power saving option that lets Windows turn it off.

Some driver versions for the Intel Wireless-AC 9560 can also cause this, so updating or rolling back the driver might help. Until Microsoft fixes it, the most reliable workaround is to use a cable or the Xbox Wireless Adapter.