I had the same problem with my thermostat. I did all the tests you described with no results, then turned the thermostat on and off several times and it fired right up.
I think the contacts on the on/off switch were dirty from lack of use. Turning off and on several times restored a good connection. It has worked fine since.
My thermostat runs off of 12 volts at all times, from the battery or through the converter. Its on/off switch is to complete a 12 volt
circuit, much like a light switch, to turn the furnace on or off.
I think your thermostat is probably OK, just needed a little exercise.