... which is still insanely wasteful because methane synthesis from scratch is just 50-60% efficient, and capacities are going to be limited - we need all the synth-fuel we can get for long distance air flight and transoceanic ships because these two usecases can't reasonably be shifted towards electric propulsion.
Making synth-methane just to burn it in a stove which loses a lot of heat to the environment compared to an induction stove.
I'd drop this argument your other points are good but while gas might technically be finite it's not on any timescale that matters here.