I don’t know about New Mexico - but they do in Nevada.
Yes, you did mean to cause the reactions - you did not post your first in good faith. You posted to elicit a squabble.
The argument made by editor-surveyor does not depend on grazing rights existing in perpetuity, it rests on the fact that the transfer to the US government made in the treaty was not one of property rights (of which grazing is one), but a transfer of governmental authority.
It would behoove you to read what others say, and understand it, before you start constructing all of your strawmen.
So where do the property rights reside as a result of the treaty? Do you know of any FReeper authorities on the matter? /s