The claims are either that:
Animal fertilizer is used because it's available as a cheap byproduct. We use it to fertilize plants for human consumption, sure, but since much of the plants we grow are fed to animals (such as 80% of all soy), most of this fertilizer is just growing plants that feed the animals that make the fertilizer.
There's a reason that we are manufacturing fertilizer by the Haber Process. If this claim were true we wouldn't need to make any additional fertilizer, since it would be a self-sustaining system. But the obvious should be apparent: You aren't somehow generating nitrates out of nowhere by feeding the plants to an animal first. The more animals you grow, the more food you need to grow to feed them, the more agricultural runoff you get, and therefore, the more fertilizer you need to manufacture to cover that loss.