CO2 vs Methane
"Methane is, of course, a vastly more powerful greenhouse gas than CO2"
True but rather misleading. Methane is not long-term persistent in the atmosphere. It oxidizes to CO2 plus water. Half-life is about twelve years. More methane is contributed by live cows than landfill. That in turn is dwarfed by what could be released if methane hydrates in permafrost started thawing out (which has the potential for a rapid runaway one-shot global warming event, which the recent fossil record suggests has happened many times in the past).
The story for CO2 is far more complex. Plants absorb a few percent of the total atmospheric CO2 every spring, but give almost all of it back come autumn. There's an annual CO2-concentration oscillation reflecting the greater land area in the Northern hemisphere temperate zone compared to the Southern hemisphere. We don't know whether the vegetable kingdom responds to more CO2 in the atmosphere and/or higher temperatures by sinking any carbon more permanently, let alone on what timescale. We do know that the more CO2 in the atmosphere, the more a plant adjusts its growth to favour roots over leaves.
We could almost certainly selectvely breed strains of wheat that encapsulate carbon in silica nodules in their roots, thereby rot-proofing it in the soil for milennia. Wheat naturally does this, some strains much more than others. Given the scale of wheat cultivation, this could provide a significant CO2 sink. AFAIK it's not known why wheat and some other grasses evolved to do this.
Most Greens don't know any of these things. It begs too many difficult questions and offers not enough certain answers.
Don't get me started on the complexities of water vapour - another greenhouse gas nearly as potent as methane, but when it condenses it forms clouds, which reflect sunlight and cool the planet.