I don't think you should get distracted. The problem of the flame going out when you let go of the knob is the thermocouple or possibly the magnetic valve it connects to. 99 time out of 100 it is the thermocouple.
If it lights with the knob pressed in, the knob is pressing in far enough. If it works on mains, the coolant is circulating.
If (when) the gas is staying lit and giving a strong blue flame, if the fridge is not getting cold, there are other things to look at.
But start by sorting out the thermocouple issue first. Well, second, after fixing the leak!
ALWAYS leak test any connections you have disturbed before considering the job done.