Is this a new issue? Are you suggesting that the new admin should be an admin and moderator? I wonder if this was a design decision long ago or a new change.
As regards chat, it seems to me like the admin user should be able to access all chat even if they are not listed in chat allowed groups, just like they can access everything else on the site.
As it happens, I was looking at the new user experience this week too and noticed that the groups page on a new site is potentially confusing for the first non-staff to show up. It’s bland, only the moderators group shows up on the list, and there are no members of the moderators group.
