Overgrazing can and does cause problems but controlled grazing can actually fight desertification and has been used:
https://www.scientificamerican.com/article/can-livestock-gra...The organic and natural farming methods used in parts of Asia - especially India, utilizes livestock for much more than just manure.
You should look up Zero Budget Natural Farming (ZBNF) by Subhash Palekar.
Quoting from this site:
https://www.civilsdaily.com/news/zero-budget-natural-farming...
The “four wheels” of ZBNF are ‘Jiwamrita’, ‘Bijamrita’, ‘Mulching’ and ‘Waaphasa’.
- Jiwamrita is a fermented mixture of cow dung and urine (of desi (Indian) breeds), jaggery, pulses flour, water and soil from the farm bund.
- This isn’t a fertiliser, but just a source of some 500 crore micro-organisms that can convert all the necessary “non-available” nutrients into “available” form.
- Bijamrita is a mix of desi cow dung and urine, water, bund soil and lime that is used as a seed treatment solution prior to sowing.
- Mulching, or covering the plants with a layer of dried straw or fallen leaves, is meant to conserve soil moisture and keep the temperature around the roots at 25-32 degrees Celsius, which allows the microorganisms to do their job.
- Waaphasa, or providing water to maintain the required moisture-air balance, also achieves the same objective.