I don't think anyone is suggesting that meat production at present doesn't add something to the system, but CO2 and methane emitted by the process isn't exactly new. Whether it's corn or grass that's being fed on, the carbon being belched out by cows comes from the carbon the plants captured from the atmosphere before being eaten. While inevitably some carbon is going to be consumed from the ground, it is minuscule compared to all the other minerals that the plants need to consume. Why would they need to get carbon from the ground when photosynthesis already necessitates using atmospheric carbon? And yes, methane eventually degrades into carbon over around a decade or so.
Also, there were far more large grass eating ruminants on planes like the North American continent, before humans killed the vast majority of them, than there are cows being raised as livestock now. We don't even have more cows being raised than a decade ago. So trying to reduce meat consumption is not only fighting against the atmosphere's own carbon cycle, but the same cycle that was occurring in nature at a greater abundance before man showed up.
We're better off just not releasing carbon that's trapped in the earth's crust, which is what we're doing when we're burning fossil fuels, because that wouldn't really be happening at such a rapid rate in nature. But no one wants to talk about that. We have nuclear, but the hand waving never stops. I don't really know what to tell people who think that we need to stop eating meat because of global warming when they refuse to advocate for nuclear energy.