Nice! I had many of the same thoughts. Except I now think that the volume expansion is not relevant, only the mass flow is. And obviously the air mass flow dwarfs the fuel mass flow in a jet engine. (I gave some numbers, just from Wiki specifications.)
The specific heat of the gases (CO2 and H2O versus N2 and O2) might be significant, however.
I would also be interested in hearing from an expert!