I think Moin has discovered a bug in the chat system. Also, I think there is a feature suggestion in here. There is no group setting to determine which members are allowed to start DM chats with specific groups. This is especially important for automatic groups like admin and moderators who likely do not want members to start DMs with them as a group.
(Let’s leave the community-moderators group aside, and also the question of whether members should be allowed to start chat DMs with the moderators group (the answer is no, as is being discussed in Moin’s Site feedback topic)