Excluding the farm equipment and fossil-fuel-based fertilizer, cow burps form a natural closed loop. Meaning, grow corn or grass and it takes in greenhouse gasses from the soil and the air. Feed it to a cow and it releases those same greenhouse gasses.
Cows and plants by themselves are not actually contributing greenhouse gasses into the atmosphere in a manner that contributes to global warming. Meaning, if you're worried about cow burps or humans breathing... stop worrying about it. And if you're pointing out that animals emit greenhouse gasses... cool, but that is irrelevant to a rational discussion of emissions that lead to climate change.
For agriculture, the system becomes imbalanced when you add in the emissions of the industrial farm equipment and the shipping infrastructure to getting food from the farm to the the table. But that imbalance is what we're trying to address when we talk about transitioning to more fuel efficient vehicles or more sustainable farming practices (like not using fossil fuels in fertilizer).
So, mentioning agriculture as a polluter is typically a red herring. At first blush it seems reasonable because there are a lot of farm animals that emit greenhouse gasses. However, including agriculture in a discussion of emissions is usually disingenuous or misinformed.