The gas flame lights the first time but fails to do so after that. The ignitor continues to cycle on and off normally. From watching your video (great video!) I would agree that it is likely a problem with the gas solenoid coils but I am curious about two things before I order the parts: 1) the two terminal solenoid measures 1300 ohms and the three terminal solenoid measures 600 ohms on the outside terminals and an open circuit from the middle terminal to either outside terminal. Are those good numbers or is the middle terminal bad? 2) I measured 0V on the solenoid terminals when the ignitor is lit and 120V when the ignitor turns off. Isn't that backwards from expected operation?
(by the way, I reinstalled the coils and now the flame never lights even though the ignitor still turns on... I guess the coils are no longer intermittent)

Well I went ahead and ordered the gas solenoid coils and that fixed the problem. Thanks for the fast shipping and the video support!