Iirc, the primary problem with gas stoves is nitrogen oxides which are created by just about any burning process that uses atmosphere as the oxygen source since air is about 78% nitrogen. In other words, the problem doesn't have to do with the methane being a fossil fuel.
If we had stoves with a source of gaseous or liquid oxygen, then the NOx emissions would be greatly reduced.
People forget your breathing in the exhaust of burning a fossil fuel.