Not super impressed with Dometic fridge controls

I have the same and it works "OK" for us. I agree that the buttons could stick out farther but I'm ok with the fact that there is no wasted energy going to energize a light when it's on gas-only because that's generally when I'm dry camping and using a genny. OTOH, I wouldn't have minded if there would have been a different color LED for each function including the gas only mode.
But yours is working the same way ours does in that the LED is not on when set to run just on gas.