The reason is simple: to produce the same amount of energy (and protein) with livestock as you would with crops, you have to use orders of magnitude more resources and you affect the environment orders of magnitude more: you use more fuel, release more greenhouse gases, use more water, use more land area, destroy more soil, et.c.
And while it might be true that one hectare used for grazing might be more biodiverse than one hectare used for, your example, soybeans. But 85 percent of the world's soybean crop is processed into meal and vegetable oil, and virtually all of that meal is used in animal feed.
I think you need to look up the meaning of the expression "straw man". And about the rainforests: Tropical rainforests are the most biologically diverse terrestrial ecosystems in the world, and right now, huge areas of that rainforests are destroyed for the land to be used to raise cattle.
> It sounds like you are suggesting non-animal agriculture kills or displaces more animals than animal agriculture.
Animal agriculture is better for biodiversity than crop monocultures. I understand that the truth can be shocking for some people, but there are dozens of ecological studies pointing to the environmental value of pastures and meadows, specially as ecotones.
As the initial plan to demonize cattle failed, now you are silently replacing "non-animal agriculture" by "rainforests" as a straw man to attack the original claim. Sorry, but this was not the point that we were discussing.
"Non-animal agriculture", also known as "Agriculture", has replaced happily as many rainforest hectares as cattle, if not more. Take a look to the fate of Indonesian rainforests