Ok how gas works
Thermisistor inside cold box gets feelin a tad warm and changes it's resistance
Control board senses this and thinks "Guess it's time to heat the boiler up" (NOTE this part is working cause it works on electric) so let's start watching her.
It opens the gas valve and starts the Direct Spark Generator.. you should hear the "CLICK" of the solenoid and the "SNAP SNAP" of the ignightor if you do not you have suspects.. IF you hear the CLICK but no snap the solenoid valve is LIKELY good (not guaranteed) if you do not hear the click measure voltage on the solenoid (Should be near battery voltage) if none control board or connection bad.
If valve working (You may be able to detect the gas smell/flow) And no SNAP look at the ignition point spacing. Compare to what the owners manual states it should be.. NOT on a fridge but I've worked on two furnaces. one house one RV with DSI both had wide spacing. Resetting made a 100% improvement.
Now if you have gas flow and spark you SHOULD have flame Look at it you may need to adjust air/fuel mix a bit... If no or poor flame the gas jet (orifice) may need attantion (Cleaning). Or there may be a supply side issue (You topped off the tank,,, Did you or the technician remember to open the valve when done?)
Now the flame heats the thermocouple (this is different from a thermistor) and it starts to generate voltage/current (Not much of eaither but enough) the control board senses this and continues gas flow till the Thermisistor inside the cold box starts feelign "Chilly" once again. at which time it shuts down.
Possible issues
No flame: Spark: Bad control bad connection Spark gap too big
No flame: No gas flow, bad vavle, tank off, other obstruction, clogged gas jet
Flame but shuts off swiftly. Thermocouple dirty... bad bad control board or connection
OVERHEAT... flu needs cleaning
That is about all I can think of but I'm sure I've missed some stuff.