Tagging PMs will be a huge boon so I hope this is coming soon.
But it’s not pressing since it’s only really an issue for moderators/staff, not regular members, luckily.
In terms of my workflow. I recently went through the whole backlog of staff messages - it took me a few hours a day for a week but I succeeded and felt good afterwards. 
Now I try to stay on top of them on a daily basis and use the ARCHIVE link liberally to get out of sight those messages that have no reply from a member or don’t need handling. I prioritize those messages that have been replied to by members.
I also make liberal use of the ASSIGN feature to assign messages to colleagues or to myself that I don’t want to deal with right now and then hit ARCHIVE.
And finally, I use the TIMER feature so I can be reminded later about messages that I want to make sure I follow up on.
Oh, I almost forgot. As part of this cleanup I invited the @moderators group to all staff messages and make sure moderators are included in future. This helps separate my personal messages from staff messages, which often need special handling, because when a group I’m in is included in a message it does not show up in my own messages inbox/archive. It’s only in the group inbox/archive.
Also, I rename personal messages all the time to add member specific details to them or to add statuses (e.g. UNDERWAY or DONE), and then search for those keywords.