As an update, I finally had some dry weather to work on the boat this weekend. I took the old cable and chain assembly out. One snag I'll have to deal with on re-installation is that only one of my messenger lines survived the journey - the second got cut up in the conduit at the bottom of the pedestal and the cable pulled free. I think I'll be able to get the second line up to the top using fish tape or a coat hanger.
The steering cable was, in fact, in terrible shape. Of the seven strands of the 7x19 cable, only two were still holding. It was definitely an accident waiting to happen. I can't believe my marina told me it didn't need to be replaced. I can only guess that they shone a flashlight from the front and didn't actually crawl under to look at it.
The failure point was right where the wire was crimped (see picture). I don't see any rudder stops in place, so I think the problem is that the quadrant is being permitted to swing so far to each side that it bends the cable at too tight of an angle. I'll have to investigate what kind of stop I can install.