In my opinion the most damaging part of the film Cowspiracy is its terrible treatment of sustainable livestock management techniques. So I have put together a bunch of evidence supporting the effectiveness of using livestock to regenerate the environment. You can access that here: "Evidence Supporting Holistic Management"
You might also be interested in these responses to the most commonly cited anti-Allan Savory articles:
There's an IndieGoGo campaign for a new documentary that talks about how ruminants are essential for recreating topsoil in brittle environments, places where for months out of the year, the only moist place is inside a ruminant's stomach(s). It seems one of the focus points of the movie is pointing out that eating meat doesn't necessarily kill the planet and going vegan doesn't necessarily save the planet.
I think the problem is in numbers eventually, in nature the number of herbivores and also any animal that feeds on plants will be controlled by the predators, so their numbers will be way lower, also grazing animals die a lot during winter and during the dry season.
What humans changed?
What humans do is they find a fertile place, they slash and burn everything, then they plant crops until the soil gets too poor for their short rooted annual crops, then they leave it to grass to take over, and feed their animals on that land in a very unsustainable numbers, the number of animals is way higher than in a natural ecosystem because humans will store hay for the winter, or in a dry area they will find and provide water, also they will eliminate the predators, so thats how you end up in a situation that is much different than what happens in nature, herbivores became a pest on that planet, the problems is in humans ofc and not the animals, just humans dont know when to stop and how much they can take from nature until they start to deteriorate the potential of the place to create and sustain life.
Thats how desertification happens.
If you dont have grazing animals eventually the vegetation will become way taller, that will protect the moisture from the sun and the wind, the plants will be able to develop deeper roots, the place will have more access to both water and minerals, there will be more growth and more organic matter which further stores water and make it easier for plants, so with the same amount of rain you get more growth. Also tall vegetation will stop the wind and will capture the dust, and part of that dust is fertility, that will be accumulated and purified with time.
Some places naturally have grass and not trees because the rain is too little ofc, but many many places have been cleared from trees and turned into pastures. There are land good for crops(the most fertile), and the rest if it has enough rain will be more morally correct to be planted with trees, sure some animals can use the grass around the trees, but their numbers should be as low as in nature, or just slightly above that, far from what humans have been practicing for hundreds of years.
Grazing patterns and how you do it may have some positive effect, but I just doubt it will change things that much, mimicking nature will mostly work if people focus on the numbers of animals, or the number of humans eating that much meat and dairy products.
Companion Planting Guide by World Permaculture Association