"Cherokee" is a common family legend in the South East, much like having Wyatt Earp's illegitimate child in the family tree in Texas, Oklahoma and Kansas.
I was always taught that the claim of having a Cherokee princess in the family tree was often used to give nativism and white supremacy more credibility through self-Indigenization, which is what helped it spread and survive to the current day. And as others have pointed out, it was also used as a way to hide race mixing. It's likely that a lot of people aren't aware of this, and just think they're sharing a fun but if family trivia.
And, as I pointed out in another comment, the Cherokee Nation has no requirement for any percentage of native ancestry, so there are a lot of people in Oklahoma and the surrounding area who are more or less white, but are legit members of the Nation under it's bylaws. Which can add some confusion to the issue.
The ship was built as simply as possible and fueled with the precise amount needed for it's weight, there was nothing else to jettison besides the young woman. The plot was intentionally structured around an impossible scenario because the editor of the magazine the story originally appeared in wanted to subvert the "engineer action hero saves the day with a clever idea" trope that was common when it was written. The heavily contrived scenario is the weak point by most people's estimation, but overall the writing is well done and characterizations are very good.
The story bugs a lot of people due to the total lack of any safety margin for such an important mission as delivering emergency medical supplies. A guy named Don Sakers even wrote a rebuttal called The Cold Solution that was meant to point out a few things the original story overlooked without the idea of a bare minimum ship being changed.