I do not think Warren comes out of this very well. 8 generations prior, the most likely ancestor is something like 200 years ago, and 1/256? I would be really surprised if the average American doesn't have more than this.
"Best" case scenario, 6 generations prior is still 1/64 (not 1/32 as reported). I dont see many contexts (including medical) where this would be considered having "native american ancestry" in practical terms. Though technically correct, it's as dumb as me saying I have African ancestry, with all members of my family until at least 4 generations prior being white. The average (white) Portuguese has something like 2 to 3% admixture with African haplotypes so, around 1/32.