This is the reason consoles do not support Bluetooth audio and wireless headsets all need a dongle.
[0] https://www.logitech.com/en-us/products/keyboards/mx-keys-mi...
There are also some BT limitations at play. Key rollover is capped, and mouse polling rate is lower. But that'd only matter to me if I were playing PC games. Ability to switch devices is a plus for BT.