I found the idea intriguing, but the argumentation does not seem to hold up.
Take someone who was born half-Cherokee in 1750, and had exactly two surviving children whose descendants each had exactly two surviving breeding children. Assume 25 years per generation.
By 1800, we have 4 1/8-Cherokee descendants
By 1900 it’s 128 1/256-Cherokee children
By 1975 it’s 1024 1/2048-Cherokee children
You say Euro-Cherokee affaires, flingsd and marriages were indeed fairly frequent. Not surprising if Cherokees were more good-looking to European eyes than, say, Sioux or Mohawk. But this means that out of group Civil-war-era Confederate patriot, maybe half have at least 1/128 Cherokee heritage, and the claims made for whatever propaganda purposes are more likely to be also true than not...