OK, I might be making some progress.  I re-flowed the solder on the lugs, but that didn't change anything.  I put the machine back into switch test mode and instead of looking for stuck switches, I manually closed every switch by hand.  Low and behold the only one that did not register as being closed was that left slingshot.  Since it does fire in the solenoid test mode, does that pretty much rule out everything but the diodes on the coil?