What is in bold is a
massive number and frankly a lot higher than I expected. One-seventh is not "statistically insignificant" in any way shape or form, it is probably comparable to entire industries put together.
Either way - from what I understood, and also from how you're framing it, the argument isn't that we're saved if we change our diets, it is that we're fecked if we don't. With the difference being that we need a transformation much wider than diet alone to be "saved". (The assumption through all this is that as the developing countries become richer, meat consumption and overall lifestyle of their newly wealthy classes becomes equivalent to the west, and that has certainly happened in China and despite cultural issues, in India too).
As
@George Owen pointed out, the greenhouse gases are one aspect of the environmental impact of meat, of which you focused on the methane emission of cattle. Expanding a little outwards from that we see rapid deforestation of the Amazon gor grazing and soy production for fodder. Both are net contrbutors to atmospheric carbon and are intimately linked to cattle farming.
Going further out, we encounter the simple biological truth of trophic levels - i.e., energy is lost at east level of consumption within a food web. Say a plant's efficiency of sunlight to carbs is 10%, the cow's efficiency of plant to meat is 10%, which means that for every unit of sunlight you get 0.1 units of plant matter and 0.01 units of animal flesh. What are the environmental implications of this? Land and water use. Fodder needed to grow animals to generate meat requires a lot more land and water than if the crops were used to feed humans directly. Same with water, and then you add the water the animals themselves need.
At this point there are 2 objections raised - one is that pasture land is not necessarily crop land, which is true, and the other is that non-factory farming doesn't need so much fodder production since you will let the animals graze.
For point one, it is true sometimes, not always, and besides the entire point of the trophic level argument is that you don't need to convert all fodder land to foodcrop land since producing crops is simply more efficient. For the second point, meat consumption at western levels requires CAFOs which are the extremely efficient energy to meat producing operations possible. Allowing all the cattle and goats of the world to graze would quickly denude massive green areas. Capitalism is good at optimising along any axes that affects profits, and CAFOs are one of the triumphs of capitalism.
Finally, the least talked-about and potentially most dangerous effect of modern efficient meat production is antibiotic resistance. Since animals can be grown quicker, with less resources like land if they're cramped together in unhygeinic conditions, antibiotic overuse to allow them to survive and grow quickly is very common in modern farms from the US to Netherlands to India. This has already created some resistant bacterial strains.
So basically we have a situation where on every metric, meat is more harmful to the envionment than a plant substitution. This has been shown in study after study, with almost all showing that vegan and vegetarian diets have the least environmental impacts. Of course this is strictly universally true - someone who lives off a single animal he shot in the wild has effetively no environmental impact. But it is impossible for 7 billion people to have that lifestyle, and on net the veg*n diets are more sustainable.
I agree that the actual best case for veg*ism is the animals case, and it is why I don't eat meat. But there is less chance of a mass public breakthrough of that message then there is of Lenin's reanimated corpse reviving the USSR. Can you expect everyne to freely choose to give themselves less of something they like?