Edit: And would this even be the solution? Methane is a greenhouse-gas after all...
The wikipedia article on methane says something like "There is little incentive to produce methane industrially". Basically, we have too much of it already, from fracking, from livestock, from all sorts of places.
Giving it to SpaceX to burn as rocket fuel makes it a less potent greenhouse gas.
Burning the methane will of course release CO2 back into the atmosphere, but if they indeed make the methane by pulling CO2 from the air, it would effectively make Starship (the fuel anyway) carbon neutral.
(I guess actually it's a bit carbon negative because some of the engine burn will happen in space).
The happy consequence is relatively low net-carbon emissions.
[0] https://cleantechnica.com/2019/10/30/no-you-dont-have-to-wor...
Methane + O2 <=> CO2 + H2O + energy
https://www.middleschoolchemistry.com/multimedia/chapter6/le...