I've never found bluetooth controllers to work that well on linux. The pairing process seems to always be flaky, and then latency issues are often an issue as well.. Certainly not seamless like using them on the switch is.
I think it depends on your computer's bluetooth hardware.
Mine reports itself as a Cambridge Silicon Radio USB device 0A12:0001, and works great with recent Sony controllers, while people with some other dongles have reported lag and/or unstable connections.
This list is a bit old, and wasn't aimed at Nintendo controllers, but might still be a good place to start:
My experience is the exact opposite, although this probably depends on the quality of the Bluetooth driver in use. My Stadia controller works great on Linux, autoconnecting on power-on and everything.
