"shorting the terminals where the thermostat wires went would open the valve"
Yes, it is a millivolt valve...it WAS faulty (the guy at the appliance parts store checked it) and needed replacing. I replaced it with an exact match for the valve I took off of it. And shorting the thermostat terminals does open the valve (it seems that it has a delay of a few seconds...I just now realized that when checking it again).
There's a thermostat that could be faulty. There's also another switch that (I assume, from what it looks like) closes when there is sufficient water pressure in the heater. Either of these could be faulty, I suppose.
"Most problems are from bad connections and wire that is too long with bad thermostat contact"
I don't believe that there are any bad connections...I've looked things over pretty well.
Wires too long? hmmm. The thermostat control and pressure switch are on the unit, so only a couple feet or so of total wire would be necessary. However, there's extra wire under there (wire to the thermostat, and to the water pressure switch), coiled up and tied with a cable tie...probably two feet extra or so. It's (approximately) 18 gauge aluminum wire.
The wires being too long would cause the unit to not function properly? The wires ARE long...and could be shortened. But I can only assume that the unit was functioning properly at some time with it wired like this...so I doubt the wiring is the culprit.