Continuing the discussion from Split into two settings: "Who can message and @mention this group?":
It seems that in order to be able to message a group you must also have to be able to @mention it. If you cannot @mention a group it won’t be recommended in the auto-complete list of users; you can, however, type the group’s whole name and be able to message the group.
Now that @mention and message are separate settings, it seems like a bug that completion on the message interface works only if you can also @mention the group.