My current solution is that I have turned the pointless field for entering the name into a feature.
This does not really solve it, because it only makes sense in cases where the admin wants to enter a custom display name for the group. Otherwise, they still only add names for being able to reorder the groups while seeing their names and need to remove the names afterwards.
The object setting showing the group’s name based on the values of the groups dropdown would still be the better solution.