> Is that process/workflow/criteria at least written down somewhere so
> someone could start hunting for a replacement without needing too much
> help from the team?

Yes, we have a slide deck that talks about it.  We could move it on-wiki if anyone's interested.  Here's a rough high level description:

Epics are high level things that our stakeholders would like us to focus on.  By stakeholders we mean the community, people in WMF, etc.
Every quarter, we prioritize the Epics that we'll focus on.
We break down prioritized Epics into Features and gather medium-level requirements there.
We then decide to do Features as part of Sprints (two week work periods).  When we do that, we break down the features into Tasks.
Features, tasks, and epics have statuses that they move through according to a few simple rules.  Features are estimated in points and that's how we figure out what we can do in a given period of time.

This is pretty standard for agile, and most freemium and for-pay tools let you implement this process.  But, out-of-the-box, Redmine, Trac, Bugzilla, etc. are not easy to beat into submission.
 
For what is worth:

https://www.mediawiki.org/wiki/Project_management_tools

http://lists.wikimedia.org/pipermail/teampractices/2013-December/000205.html

Personally, I'd love to move to something like github + gitban or Phabricator.  I value user experience over the hacker ethics but Mingle kind of fails me on both :)  However, we've invested a lot into our process and there were some very good reasons for that.  So, for now, I think we have bigger and simpler fish to fry.

Hope that helps, but feel free to ask anything if you're interested.