To be more specific, the cows should take a bite and move on. This keeps the grass short, but doesn't kill it. With short grass, predators can see prey easier, so birds can eat and deposit droppings to add more nitrogen to the soil.
We can simulate buffalo mob grazing with "managed intensive grazing": many cows in a small area for about a day, then they move. In addition to what you said, it manures the land quite evenly. The grass can stay in its fastest-growing "adolescent" stage. Builds topsoil quickly and sequesters unbelievable amounts of carbon. (This is not how most beef is raised.)
Unfortunately, if we were to take all the cows stuck in factory farms and turn them loose on BLM land, I somehow doubt there would be enough food out there for them.
Grass-feed cattle farmers tell me it takes almost exactly the same amount of land to graze cows as it does to grow the grain to feed them in a feedlot, so that would be a direction to look: replacing grain farming with ecological grazing.
