I am realizing I cannot convince you that grazing has an ecological benefit. I will leave you with this final information.
Mowing and grazing have different outcomes. Below is a link that talks about the use of patch burn grazing and the results.
Here is a link that shows a result that is being attributed to annual mowing for hay.
Here is a book that covers a number of important studies on Prairie Ecology.
There are a number of other books I have not yet read that I sure would also be enlightening.
Regarding the whole humans not being a part of nature issue. These grasslands were rotationally burned by Native Americans to attract Bison since these ecosystems developed at the end of the last ice age. Ranchers have continued this management regime with cattle. Indeed, a number of ranches are the best remaining example of this type of ecosystem and cultural tradition. You are correct that cattle are different than bison. They have small differences in grazing patterns. Their dung is less completely digested. I am sure there are other factors too. However, in instances where hard to contain-dangerous bison cannot be used, cattle make the best substitute. It has been shown that active management (fire, grazing, invasive species control) must occur if this ecosystem and all the species it contains are to be preserved. Even if this does not fit into your definition of "nature", then I still believe these activities are worth doing from the stand point of species preservation.