This is only true when there is plenty of grass available. In fact, you're wholly incorrect because of the extent to which we've used up this planet as an input for meat production. We've essentially turned
all of the available arable land into farmland, and
all of the available grassland into pasture. There is nothing left but forest, and we are quickly replacing that to our great detriment.
We are tapped out. If the rest of the planet starts eating meat at the rate that the first world does (~150 kg of beef per person per year), we'll need another planet to feed them. There is literally not enough land to do it on.
I recommend this XKCD for an idea of the scale to which we've appropriated Earth for raising our livestock:
https://xkcd.com/1338/
And this UN FAO report, "Livestock's long shadow", on how much of Earth we've actually used for raising livestock, and the devastating environmental consequences:
http://www.fao.org/docrep/010/a0701e/a0701e00.htm
edit: adding this figure from the report about land usage just so you understand the situation clearly:
http://i.imgur.com/rvbCwA0.png