I thought of another reason you may not have flame after the button is let up.
The older units, like yours, had a 2 level flame. The high level is the one which does cooling when the thermostat calls for it, otherwise the flame falls back to the lower level. There is no pilot light on those refrigerators, only high flame and low flame.
So, high is controlled by the thermostat, low is fed by the "bypass" screw opening. If the thermostat is at it's lowest position and not calling for high flame, and the bypass opening is plugged the flame will go out when you let go of the reset button.
Try this: Be sure the cold control is turned up as high as it will go, then try lighting the unit. If it lights, but then goes out when things cool down, you need to clean the bypass opening. It has been a very long time since I have done that, but as I recall it is a little brass head nut like thing on the side of the gas valve the thermocouple goes into. Or, if you have instructions, they may identify it.
So, the gas for low flame goes through the bypass opening, and for high flame it goes through the thermostatically controlled passageway. If the thermostat says it is cold enough or is in the off position, and the bypass opening is plugged you will not be able to keep the flame on.
I hope that helps if the thermocouple ideas do not.