NPS generally doesn't allow grazing other than probably Capitol Reef which was made from existing BLM land with existing leases. Organ Pipe isn't a good example as the Sonoran Desert probably isn't a good candidate to reverse desertification and that ecoregion is important in itself.

When trying to reverse desertification, herd animals aren't the most important thing, stopping soil erosion is. Grazing animals can also be a negative, but if managed properly they can be a great positive. Ruminant animals are an important part of the ecosystem, whether they be bovine (including bison), ovine, caprine or deer. I strongly suggest you watch this Ted talk. It's certainly not settled science, as some will argue that the benefit is offset by the methane created by ruminant animals. I would argue that the methane production from the great bison herds of North America, which were also ruminant, didn't seem to be a problem, or we wouldn't call climate change man-made.