The way that many people eat meat is not natural at all, and these unnatural food practices are the biggest threats to the climate. The way that humans eat meat in the twenty-first century is vastly different than the way that primitive humans ate meat. Factory farms, which are where 99% of meat in the United States comes from,
are completely artificial human constructs that have no basis in natural meat consumption. Animals are kept in extremely close quarters in foul, unhygienic, cruel cages and are fed unnatural diets that are packed with antibiotics because the animals would fall ill from their horrible living conditions if they weren't fed antibiotics. Even the animals themselves are much different than the wild animals that primitive humans ate and kept, even once primitive humans domesticated livestock. The animals that we eat today have been selectively bred and genetically altered for decades to grow freakishly fast and to pack on more fat and muscle than wild animals. Primitive humans killed or scavenged the meat that they ate, with that meat coming from wild animals. There is nothing natural about factory farming, which is the biggest offender when it comes to pollution from meat production.
Also, just because something is "natural" doesn't mean that we should continue to practice it in the twenty-first century. It's also natural for humans to forage for the plants that they eat and to not wear shoes, but we don't follow those practices. On the other hand, using modern technology, like phones and toilets, is completely unnatural, but that doesn't mean we shouldn't use it. Artificial coloring in food, heavily processed food, microwave meals, eating off of plates, grocery stores, and cheese are all unnatural foods/food practices that have become part of our every day life. There is no reason to eat factory farm meat, something extremely unnatural, because humans in the past ate meat from wild animals.
One could also point out that arguing for something because "it's natural," when the consequence is climate change, may not be something worth arguing for.