Turned out, migrated supervised accounts need their XBox profile tweaked to allow to connect to the Minecraft servers.
Direct link is
Replace "Kid'sNickname" with actual nickname of your kid and change these two permissions:
- "You can play with people outside of Xbox Live" - Allow
- "You can join multiplayer games" - Allow
Hit "submit" and wait for page reload (it's SLOW). On next login, your kid should join the server just fine.