I wonder if anyone else has stumbled across something similar and knows how to fix:
The option to Add a new group is missing from the entire site.
I first noticed it wouldn't show in the "Add new..." block and thought it's a permission issue.
In the end, I checked and it is missing throughout the site, for all users, including the Administrator.
Accessing the URL /page.php?i=create-group-profile works and I can create new groups, and permissions work appropriately (i.e. levels without "Create Group" get the correct message displayed.
I uninstalled/reinstalled Groups, cleared the cache, cleared the browser cache, no success.
Interestingly enough, I couldn't find the option to start a new Discussion here on una.io as well. Not sure if related but strange enough - as I had to use the same method (direct URL in browser address field)