If she brought it up during her application to teach at Harvard, then she probably was trying to benefit from a perceived diversity preference. No hiring authority is ever going to say that she was hired because of it, and it probably wouldn't even be true if they did; they would accurately classify it as one of a number of factors that went into the decision.
But let's be honest: she didn't put down Native American on stuff because she was proud of that obviously tiny bit of ancestry. She put it down to gain an advantage. Whether it played out that way is impossible to prove one way or another, but the only way to put this behind her is to admit that she did it to try to advance herself in some way, and apologize for that. It might kill her politically, or she might be forgiven, but it's the only way to get past it. It would certainly be a refreshing change from the self-aggrandizement of our current president, and maybe that would be enough or maybe it wouldn't, but if she ties it to her story - humble background, made it pretty far in life on smarts, competence and determination, then ties that to her policy prescriptions designed to help others do the same, it might help. This is what she's done sort of circuitously, but she needs to make it an explicit narrative. It still won't go away, but at some point people will get tired of hearing about it and it may lose its punch.