Let me explain the system to you: grass is pretty and it's green; cows, sheep, and horses eat the pretty, green grass. Generally speaking, we then eat the cows and sheep, and on the rare occasion, the horses.
Now, it might seem more efficient to simply bypass the cow and sheep, and go straight to eating the grass. The problem is as far as I can tell every culture who went to eating the grass was wiped out by the cultures who continued to eat grass-eaters.
Relayed by America's 1st Sgt. over at Castra Praetoria