This is all very interesting - thanks. On my end, as stated in the OP, I have hesitated to roll out group inboxes more widely because the functionality is confusing and creates messiness. It’s not clear when to use it and how members can manage the messages. Mostly this comes down to the fact that messages are actually topics refactored to only be visible to individual members included in the message. It’s weird that members can’t delete their messages, for example, until you understand that they are actually topics that exist only once in the system. If I were to be able to delete it for myself, I’d delete it for everyone else! 
This confusion and messiness is real and disruptive both for staff (who should know better as experts of discourse) and for members (ordinary civilians trying to go about their every day lives who we try to serve with this community). I wrote up another post on this to try to clarify the situation for my staff - take a look and see if it matches up with your own experience. I’d be interested to hear from you, and happy to see the topic wikified to add more use cases.
Honestly, with this particular topic I was hoping we could find a way to cut through the noise for ordinary civilians and I still think it’s a good (the best?) idea. I am not excited about opening a pandora’s box with a range of different kinds of inboxes for messages that I then have to explain to my members. When in doubt, why not just use a secure category?
By simply putting all messages in the inbox for everyone by default, I think we will erase most confusion about messages. They all land in the inbox whether they were addressed to me directly or to a group I am in. I can then archive it and it goes into my own archive. Messages I have contributed to also appear in my sent folder. When tagging of messages comes along, I will then be able to organize my messages as I see fit like gmail labels.
If I am in a group used by a team to provide support (by email and PM) and don’t want to see all the group messages in my own messages, I can turn on the group inbox to separate out those messages.
Potential problems that occur to me in my pre-
state:
- if I am kicked out of a group, do messages I’ve already received and replied to still appear in my inbox, archive and sent folders (and tagged folders when that is implemented)? Or do I lose access to those messages completely when I leave the group? confusing.
- if I don’t use group inbox but others in my group do, if they archive a message does it archive it for me? confusing.
- later, when tagged messages is implemented, do I see tags for group messages created by others? do I lose those tags when I leave a group?
Given these potential pitfalls, maybe the answer here is to have a group option for admins to “enable team inbox for messages” and otherwise just have everyone’s messages land in their inbox as they would expect. For the few of us who need a group inbox for team usage we’d see those folders in our inbox and then not be surprised to lose access to the folders and messages we haven’t contributed to (and their tags) when someday our role changes and we are removed from the group used for team communication. It would make sense to keep access to the messages we wrote or replied to, and in specific oddball cases where it’s required another team member can remove us from selected messages.