for example, why can’t I add a picture of a group, make a description of it, etc.?
Because it’s not a group in the same sense, and can’t be treated as such.
Moderator and admin are built in statuses for staff.
Custom descriptions for automatic groups (admin/moderator/trust levels) will be available soon along with a new groups directory design.
I don’t think there’s a reason to not have flair for those groups, there’s just a bit of a complication where flair is currently used to show a user’s primary group, and automatic groups can’t be set as primary groups… so it’s a little more complicated than it seems on the surface.
One way to get around it is to create a new group and add your moderators to that, add the flair, and make it their primary group (you can set the default moderator group’s visibility so normal users don’t see the group, if you want to avoid that redundancy). That’s what we’ve done here on Meta with our Team group.