This sounds like more of a need to adhere to a process for handling these messages than a need for a new feature.
We use group inboxes here to handle support for our hosting customers and we archive when a message has been handled. That message will pop back into the inbox of there’s another reply, though.
I have a few thoughts that might help —
- archived messages will still show as unread for any user who has not read them. A possible workflow is to have users browse the archive for unread messages.
- some of us here want to keep up on all the message history, and one way we do that is through setting the group inbox to Watching. This will notify you of every message added to those topics. Then each user can use their notifications tab in their user profile to catch up on unread topics.
You could also explore using a private category as well like @tobiaseigen suggested.
Hope that helps!