One other thing to mention is that the speed of sound of the exhaust matters quite a lot. Given the same area ratio nozzle and same gamma in the gas, the exhaust mach number is constant; a higher speed of sound thus yields a higher exhaust velocity.
My understanding is this effect is a re-framing of what I described: for a similar temperature and gamma, a lower molecular weight (or specific heat) will result in a higher speed of sound (or exit velocity).
However, I feel like this framing fails to provide a good intuition for the underlying mechanism. At the limit, Te→0K,Ma→∞anyways, so it's harder (for me at least) to understand how gas properties relate to sonic properties. Yes, holding other things constant, a lower molecular weight increases the speed of sound. But crucially, it means there's more kinetic energy to be extracted to start with.