The effects of over grazing are well known and well documented. they are even visible from Space .
Ok, so remembering some history of the American west, with the cattle ranchers fighting with sheep ranchers, the general thought was that cattle do not decimate the land the way sheep do. Horses in this are pretty much the same as cattle. The reason given for this, is that the lips of cattle (and horses) are much thicker than that of sheep, so grazed lands are not nibbled so close to the earth by cows as by sheep.
Another thing about cows and horses is that they do not fully digest what they eat - so the end product which results, still contain viable plant seed within. Roaming cattle surely eats the grass, but then deposits fertilizer AND seed as it passes.
The argument between the different ranchers in the US had far more to do with the cows being unable to eat after the sheep on shared land (some land in the west was leased grazing and some open as well as others which belonged to this or that ranch), while the sheep could still eat off the same land as the cows. This meant that the cattle ranchers were leery of their stock not having enough to eat. Nothing to do with any of the ranchers having a clue about "desertification".
Meanwhile, back to the topic at hand - while "experts" may have said for years and years that grazing / overgrazing causes desertification, there on that TED talk you see just such an expert stating that they were wrong and have been all along. I can see that OVERgrazing would cause a problem, and that keeping the animals from following a natural pattern following rains, which is what humans have done. Fenced in lands keeping animals penned in to particular grazing lands while also irrigating lands which force unnatural cycle. With this in mind AND keeping that TED talk in mind, I'd think the answer would be to force ranches to allow more movement, which means taking down a lot of fences in the American west.
On the other hand though, I think the photo from space could be misleading, at least a bit, particularly wrt what it's showing in the US. YES there are large tracts of desert there, but also, some that appears to be desert in the photo is area I know to be farm land. Depending on the season that photo was taken, could very well be simply plowed under as far as the eye can see, much like during growing season it would have corn or alfalfa (as far as the eye can see).
Regardless of the message, regardless of whether his science and postulates are correct or not, Savory captivates the attention of his audience.
Notice that he gently educates his public using simplified terms and adopts the characteristics of a teacher. He uses his education sparingly as not to overwhelm his public, yet he injects it at sensitive moments throughout his oration to support the logic of his debate. In other words: intrinsic credibility to support the logic.
His brilliance is his ability to trigger the feeling of personal responsibility and accountability from his listeners by his use of the words: “I”, “you” and then gradually “we” and “our”.
He also appeals to the spectators’ emotions by the use of dramatic, intense, vivid adjectives and nouns thus evoking fear, concern, hope and reassurance.
By the end of his address, the listeners identify not only with the subject of “desertification” but also with Savory’s trials and misfortunes as if these issues were also their very own.
Though he supports his hypotheses with logic, emotion is the strongest element of his discourse and it tends to overshadow the validity of the arguments by sparking a strong response and support.
The result: the audience is his.
On top of which, and let's face it, he did offer rather substantial proof that his methods, work. It's not the aim of nature to let the Earth's face be open to the hardships of climate. Any of those of us who have gardened know that rule number one is to keep the soil in good condition by protecting it from nakedness.
Nature's way of doing this is by covering itself with weeds. Those things we hate and destroy with burning poisons. As a result of our straight-line, 'professional' agriculture and gardening we have rid ourselves of many beautiful things and are just about completing the circle by killing off all our bees.
And all the rest actually.
IIRC in South Africa they had some success in regenerating aquifers that were no longer running by stripping out the alien invading plants. Also by restricting land load of herds in veldt areas, again IIRC rotating around fenced areas, 4 head of sheep per hectare, 1/2 head of cattle per hectare. I am sure someone could look up the actual values
Finally in terms of regeneration of exhausted savanna land, local farmers used to pull all the herds out, and then plant bags made of fertiliser, soil and a small section of veldt grass spread out at 1/ 500mm in squares. This seemed to work to repopulate soil binding grass into the areas treated.