Switch Issue as well here.
I select login, and select Save to Microsoft account (the other option to save locally just loops back). I then have to use another device to go to aka.ms/remoteconnect and enter the code.
I get a login successful email confirmation from Microsoft, from 15-30 seconds or so later I'm back at the main screen of Minecraft (where you can select Play or Settings) but then the Microsoft account gets kicked out about 10 seconds after that.
Subsequent logins work the same way, authenticate with the code, logged in, then booted.
If I clear account data under settings>profile I still have to exit the game, and relaunch. one or two tries after that it sticks. Alternatively I can clear the saved profile data, exit Minecraft, hard reboot the switch and then it will work first attempt. Hard rebooting takes forever, likely not good for the switch either, and my son cannot do any of the above on his own which really makes it difficult.
Thanks and hope that helps! Really frustrating as it takes a really long time for the game to load as well lately but that might need another topic 🙂
What's your fb