I have the opposite impression, that the grazing of cows on this land is beneficial as it mimics the role the buffalo formerly played.

If and only if it's done correctly. Both the above statement and yours are true, it really depends a lot on how it's done.

In most cases it's not done well.

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.)

"This is not how most beef is raised."

It's how most beef should be raised, though, IMO.

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.

I think that the point was that only minority of this land is really used for this purpose.

No, the point was that using such land for grazing does not actually produce a high percentage of cattle raised for meat, vs. feedlots where cattle eat corn grown on farmland.

From an ecosystem impact perspective, though, arid land that only occasionally hosts herds of cattle is still going to be significantly affected, compared to similar land that is protected from any commercial grazing.

Isn't that true of pretty much any industrial farming even plants?