The response you always get, as many comments here, is that the major environmental damage from cows comes from cutting down forests or the increased agricultural input to feed them. But they stick to "don't eat meat" because it empowers them as individuals, allows a single point of focus, and, usually, plays into their pre-conceived beliefs. That's a shame, for two reasons.

First, not eating meat is a non-starter. It's not a solution because it will never happen. Period. An omnivorous species will not stop eating meat for any reason.

Second, it's a feint. If we eliminated all cows today, we would still have our tilling, petrochemical laced agriculture and the growing human population that eats it, meaning we'd still be cutting down trees at some point to feed ourselves.

You're right. Biogenic methane is a closed loop. Not a place to look for solutions to climate change.