Actually, this is not the real problem. It's not the herding of animals that's creates the overgrazing. It's the large scale killing of predators. This creates huge herds of grazers that absolutely destroy the landscape. This is how the savannah and the great plains where created. To protect their herd and children; natives killed the lions, saber tooths, wolves, hyenas, big cats and bears. In effect creating the great grassy plains. Not saying that humans created all the large grassy plains; just saying they have become substantially bigger since humans are around.

In fact we have almost no idea what the human population was, in most areas of the world. And, none at all in the millions of square miles of the Sunda region, now under water from Java to Viet Nam, and connected to more land where the entirety of the South China Sea and Yellow Sea now are.

For many decades anthropologists and archaeologists earnestly insisted the Americas were barely populated, pre-Columbus. Now we have found major earthworks projects in both North and South America that required societies in, at minimum, tens of millions to have built and maintained.

We know appallingly little about the recent history of our species beyond ~6000 ya, with only tiny, isolated digs at older sites.

A major earthwork in Java, Gunung Padang (previously assumed to be a natural feature because of it size), dated to 22 kya has had excavations halted since 2014.

I would need to see a source for this, because it doesn't sound plausible to me. Populations were lower of course, but look at how much of the population is concentrated in city centers today.

I would hardly be surprised if rural population density was significantly higher than in the past, with more farms as well as more grazing herds, even if they were much smaller than what we have today.

But I also don't really know if this squares with the sheer amount of meat and dairy that modern humans consume. How much does factory farming offset the need for open grazing land?

Basically, this is a subject area where intelligent clever people get themselves into trouble by thinking they know more than they do. I'd rather see what the experts say.