When I tested this here on meta, by creating a tl0 user, I was able to navigate to the support-teams group page and select the Message button to send the group a message. Once I posted the message, it landed in my sent messages folder. So that is working though maybe a bit too buried for some cases. You can always add links to the hamburger menu as you see fit to suit the purposes of your site, or in a banner on your homepage etc. So I don’t see that there is much more that needs to be done here?
The Kabissa inbox in that screenshot is just shown to people in the Kabissa group. People messaging Kabissa would see the messages in their own messages.