It's not the input. It's the output. Stock animals emit gasses (like any other animal) and it's non-neglectable footprint.
It's the main reason beef and lamb are so much worse in terms of CO2-equivalent than pork or chicken, and cheese isn't good either.
"Grass-fed" doesn't scale to the meat demand. It's a complacent luxury for some people, and that's all it's ever going to be until the Human population at least halves or becomes mostly vegan (so some people can enjoy better meat).