Gas chromatography flame ionization detector (FID) - why hydrogen gas? Why is $\ce{H2}$ used in an FID, apart from the fact its combustion does not contaminate the flame?

In other words, is the temperature of the flame important or critical? Is the stability of the temperature?
The key to the answer is understanding how FID works.

The hydrogen flame has a minimal flame ionisation, what is needed for the low signal baseline.

Incoming organic molecules from the HPGC column create in the flame a lot of ions and increase the flame electric conductivity.

Using alternatives causing higher ionisation would decrease FID sensitivity that is not great even at ideal conditions, compared to e.g. ECD.