Yes, but
1) What are you actually getting in a song that speaks to those things specifically? Is it that you see the word "Shenandoah" (some authors' speculations of how to rationalize utterance they could make no sense of, like "Shanadore") so you go to the Oneida chief Shenandoah and just go with whatever thing you can imagine a guy could do or have? How does a chief of the Oneida people, of upstate New York, get connected to the Missouri River (on the border of which lies "the redskin camp"). Another chanty says, "Sally Brown, I love your daughter." Does that mean there's a history of a Sally Brown that is being told?
2) Do you mean just the version offered by Capt. Whall or the song as understood from the many and varied sources? Whale's version is just a drop in the pail compared to the many other different sources, with no trace of any "redskins." And what sense is there to imagine the song traveling down the river to the sea if there is only documentation of the songs at sea? It seems like a lot of round about work just to get at a fancy story for a song, for what purpose?

It seems more likely that Capt. Whall put down this version of the song, though his narration of it is puzzling indeed. Puzzling why? Simply because it is such an outlier among the other evidence, and because nothing else yet located supports it. So is it true? Who knows. But we do know something about Capt. Whall. We know he was a British sea captain and not, like his contemporaries like Frank Bullen, an ordinary seaman who sang chanties. We know (from his own explanation in his text) that he considered things from African American culture to be degraded and he avoided including songs in his collection if he thought they were connected to that culture. (Luckily for us, he didn't know how many songs really were connected.) We know that he made up fanciful stories or over-speculated about his material. His collection is the source of several of the legends about chanties. And we know that subsequent authors made great use of Whall's book. So both Whall's idea of Indian chiefs and his set of lyrics were replicated over and over again to form a bloc seeming to represent this "redskin." Historically, there is no bloc; there is only Whall's one entry.

People have seen this floating revival version of "Shenandoah" of Whall and woven a story around the Indian chief thing, rationalizing each bit to fit the pre-formed narrative. So I ask you to take it back to the beginning, before a narrative was formed that made it into the chap books and folk clubs and wikipedias. What if no one really sang the word "Shenandoah"? Whall himself says that "Rolling River" is an American song and that Americans sang "Shannadore." What if Whall merely assumed the "correct" thing must be "Shenandoah," in order to give it an easy explanation?