Note: I work at a startup working on removing methane from the atmosphere.

Responding to several comments: yes, we need to get rid of the excess CO2. In fact when CH4 is broken down (mostly via hydroxyl† or, over oceans, Cl reactions) you end up with CO2 anyway.

So, as you ask, why worry about CH4?

The radiative forcing function of CH4 is, depending on your baseline, 30-80X‡ worse than C02. The rise in methane levels from 750 ppbv to 1900 ppbv is believed to have raised the temperature by as much as 0.5 C. So eliminating it buys more time to work on other problems.

The second problem is you can't really eliminate* emissions -- they come from diverse sources, almost all of them natural. You can see this in that the 750 ppbv level that was reasonably flat before the widespread adoption of the steam engine (around 1775 when Watt and Boulton got their patent)** -- emission and oxidation/removal were roughly balanced. Another way to look at it is that 1900 ppbv today was all emitted over the last decade, whether "naturally" or not.

I put "naturally" in quotation marks because yes, some would be emitted anyway without humans, but most of the emissions that come from humans are from biological activity that would not otherwise have occurred: primarily agriculture, but also fossil fuel extraction. If we somehow stopped all the latter it would cut anthropogenic emission by perhaps a third; livestock maybe the same amount, or maybe half in aggregate. But you aren't going to stop growing crops or dumping stuff in landfill.

And many of these sources are quite diffuse; oilfields result in lots of seepage, and who can fit a catalytic converter to a swamp?

And of course methane bursts from melting tundra heating continental shelves and the like -- not sure anybody has an idea how much of that there is. You can see craters from these bursts if you walk around in the arctic. And as the article mentions, it's not just the frozen methane, but as the frozen tundra melts decomposition accelerates. One can imagine a significantly large offshore burst that could wipe out agriculture for a few years -- probably a bad thing if you are a fan of vertebrates.

If you want to know what human activity bugs me the most, its not livestock, it's a transition to natural gas (methane, basically) as a "greener" fuel for transport, heating and electricity generation. More gas means more leaks and emissions and further cooking of the planet. And when those ships have been launched and power plants built they'll be in operation for decades because of that's how long they'll take to depreciate. The EU has come out in favor of this; the IMO is encouraging shipping to make the transition, and of course some countries depend on its export.

PS: everyone makes cow fart jokes but a point of trivia: it's actually cow burps as the cow is a ruminant.

† For unknown reasons OH levels last year seem to have dropped. Eek!

‡ These calculations are arcane, depending on what parameters you're trying to compare. We just use 30X in the company.

* There are various projects to eradicate point sources: barns, landfills, some old mines. We need more of that too.

** OK, obviously some coal was burnt before the Boulton and Watt engine and adoption was not instantaneous but you can see the bend right around the end of the 18th century.

Edit: add the point about gas being considered a "green" fuel.