I feel silly for even posting this, but when I open the EMC, go to Organization Configuration/ Mailbox/ Database Management -> click on one of the databases, I see the member servers and their database copy status.
Now, I go to View/ Add Remove Columns... and I add in the Content Index State column. When I close the EMC and reopen it, I have to do it all over again!
How do I keep this setting?
Okay, so I figured it out. When closing the EMC, I was getting "You must close all dialog boxes before you can close Exchange Management Console" and having to forcefully close the EMC. Not realizing that this would undo my changes. There is a fix for this:
http://support.microsoft.com/kb/2618444
Edit: The latest cumulative security update for Internet Explorer fixes this issue. The fix above is superseded by this.
Can confirm it works.