I note there's been a lot of discussion about how flagging could be used for this. Perhaps it might be easier to say, how would this differ from the current Flag feature?

Ticket flagging assigns the ticket to that user, displays at the top of their ticket list, is configurable to only show up in that users ticket list, and sends the admin user an email notification both when it is initially flagged and any time it is replied to. How would this differ?