I was thinking of placing the bulb in parallel to the gas valve, not in parallel to the thermostat. So, one lead of the test light to the valve side of the thermostat, the other lead to chassis ground.I'd fear that the bulb might draw enough current to hold the gas valve open. I doubt it would open the valve, but once the valve is open it might not close. That could cause an overheating condition. Then I'd be replacing the fusible link.
But really the voltage measurement - with meter in parallel to the valve as suggested for the test light - should be fine. I can't think of a mechanism by which you'd get a misleading "ghost" voltage there, one that would be revealed as a ghost by the test light.
Have you checked continuity on your valve solenoid coils? I just replaced one of mine yesterday. It showed no continuity at room temperature and that prevented the flame from starting at all. I've heard that these coils can fail to open when they get hot, and that could cause the flame to go out early. That's much harder to diagnose. Good news is that a new pair of coils in only ~$8.