I did some more investigation this evening. I also created a little LTSpice simulation to better understand how the heater circuit works.
Anyway, one of the things I tried tonight was to place a diode across the SCR anode/cathode, effectively removing it from the circuit. While I did this, I measured the voltage across the thermostat (terminals 9 and 11 on the the heater connection on the bottom of the PCB). The thermostat is "on" (shorted) when the meter starts up, but it never goes off. Also, the little heater tube never felt like it was getting warm at all.
So this prompted me to measure the resistance of the heater by taking a measurement across terminals 9 and 10. OPEN CIRCUIT. So I think the heater may be burned out. I also suspect the SCR is not good, but will follow up on the heater first.
BTW - does anyone know approximately what resistance I should see across the two heater terminals?