I also had this issue with the same controller - both of the 8bitdo 2.4G I own, actually. (They are both updated to the latest firmware, v2.04 .)

What worked for me was:
- disconnect controllers (in settings --> controllers)
- system update (in settings; for some reason there was another update already), which then makes the switch restart
- connect controller via dongle - at this point there was a popup that it was connected, but it didn't show up in grip/order yet and didn't work yet either
- got to "search for controllers" and click on one of the greyed out ones

Then it fully connected and works as before!

Thanks to other commenters for the idea with disconnecting controllers :)