I suspect that the real reason that the Paiute's do not want the mummy DNA tested is because if it is Caucasoid, as is "Kennewick Man" it provides another data point showing that the Indians were NOT the first inhabitants of the North American Continent. So, the real reason is ethnic politics.
Of course there will be narrow parochial assertions made by the "White is Right" crowd as though this proves anything beyond the established evidence suggesting that North America has been inhabited much longer than the Lamestream Archaeologists wish to put forth and by more than one group having disparate geographical origins. Both groups are likely to have hissy fits.