@codinghorror asked me to post an update here. Short version: I still think we should add a site setting allowing moderators to create groups.
Longer version:
Since I first wrote this (wow! over two years ago) groups have evolved quite a bit. Groups have been moved out of the admin section and now exist solely in /groups (perhaps this doing this securely less technically tricky?). Groups can be freely joined by users, or they can support “join requests”. The concept of “Group Owner” allows non-staff users to manage membership of a group, as well as group settings like name, description, avatar flair, and interaction. I’ve also seen first hand how groups can be used to manage incoming emails (like we do here at Discourse).
My original thought still stands - you shouldn’t need to be an admin to create a group. I recognize that some sites wouldn’t want moderators to have access to this, so I believe a site setting makes sense. Being an admin gives users significant control over the site, including downloading a complete backup, changing themes, and even changing site settings like login. You should be able to create a group without gaining access to all those abilities.