As we all know livestock can cause severe problems in terms of soil compaction, loss of biodiversity, soil erosion and degradation, etc., but there could also be some advantages after all when it is very carefully managed. I am sceptical of most listed benefits, like increased biodiversity, but I'll describe a particular farming system that could be beneficial.
That is the mixed farming system where livestock and agricultural crops are managed in a single system where livestock feed on crop residue (stalks after harvesting) and manure is used for field fertilisation. Fertilising soil for sustained crop production is actually a real problem. In many traditional farming practices, the balance between livestock and crops meant that these two processes supported each other. Also animals can also be used to plough fields, reducing the need for imported machinery and use of fossil fuels. When crop production outstrips livestock production, fertilizers must be bought which can cause nutrient overloading and pollution. Crop production is also a more intensive use of the land, whereas grazing can support a greater range of life.
I don't really understand it all that well, but I do realise that many people's livelihoods depend on livestock and perhaps livestock can help with sustainable crop production.