I understand the presumed issue would be users sending messages as staff to other users would be unable to see those messages themselves or see responses to those messages.
However, if an initial DM is sent to a user the system could seamlessly add the staff alias allowed groups to which the sending user belongs as recipient(s) to the DM.
Similarly, if users send a DM to the staff alias username the system could add all (or a subset of) staff alias allowed groups to the message, giving users one point of contact.