After inspecting the switch itself it appears to be either melted or just worn down. Two spring mounted pegs seem to be responsible for depressing two little metal strips inside to make a connection. With the pegs worn down as they are they must not be pressing the little strips down enough to keep a connection in the third position.

I'm wondering if this is a common problem and/or if these pegs have been melted or if they are just worn down from rubbing up on the strip for 30 years. Should I just replace the switch or do I have an underlying electrical problem that I should be looking for? It certainly looks like it's been melted, and I noticed that the little block that retains the wires that the switch connects to has a little portion that is scorched at one of the wires.