For a large portion of the global Methane pollution from cattle, the carbon cycle started out with natural gas being used to produce artificial fertilizer. If we wanted to address methane pollution it would make sense to start there in order to address both leaks and the introduction of carbon into the system, but this is where politics comes in. Artificial fertilizer is not just the building block for most cattle farmers, it is also the building block for most meat alternatives and the so called "renewable and carbon free" biomass industry.
On my free time I often spend time diving in the Baltic sea, and every time I go below the surface I see the effect that artificial fertilizer has. The excess nutrient is killing the whole area, which in turn release more methane from the ocean bottom. Currently the area is around of 60,000 km2, but the effect can easily be seen in nearby "healthy" areas.