It is a closed carbon loop as long as the grass the goats turn into methane ultimately regrows. It's true that you have a sustained amount of methane due to the

It is a closed carbon loop as long as the grass the goats turn into methane ultimately regrows. It's true that you have a sustained amount of methane due to the flux through this loop. The same is true for fossil fuels, in principle the loop is closed because eventually plants will turn the CO2 back into fossil fuels. It's just that because the time constant of that loop is thousands of years and we burn it much faster than that, most of the fossil carbon will eventually make it into the atmosphere. Even though methane is a more efficient greenhouse gas than CO2, the time constant on the CO2 cycle more than makes up for that difference. When making these calculations, you can't just look at the amount of emission, you need to look at what sustained amount of atmospheric level results from that emission.