After doing a bit more digging, the reality is more interesting than the video implies. Neanderthals didn't get a rebrand because white people found out they shared DNA; the bad reputation was always built on dodgy science that was already being dismantled.
The whole "brutish caveman" thing traces back to 1908, when a French palaeontologist called Marcellin Boule got his hands on a nearly complete Neanderthal skeleton and reconstructed it as a stooped, bent-kneed, gorilla-like creature[0]. Problem is, he got it completely wrong; the skeleton had severe arthritis, which he either missed or ignored, and he projected Victorian-era ideas about racial hierarchy onto the bones. It took until the 1950s for anyone to seriously challenge it[1], by which point the image was baked into popular culture.
The thing is though that the prejudice didn't even start with Boule. From the very first specimen found in 1856[2], scientists were already calling Neanderthals primitive because 19th-century science was obsessed with ranking humans into racial categories. Neanderthals were useful as a "below us" rung on a ladder that was already bullshit. So yeah, the video's not wrong that racial ideology played a role, but framing it as "white people discovered shared DNA and then rebranded Neanderthals" is a bit too neat. The rehabilitation started decades before the DNA findings, because the racial hierarchy framework that created the caricature fell apart first.
What we actually know now[3] is that they used pigments and art, made tools, cared for their sick, buried their dead, and survived wildly different climates for hundreds of thousands of years. They weren't H. Sapiens' thick cousins; they were a genuinely capable parallel branch of humanity that we happened to absorb (and probably helped push out, though the exact mechanism is still debated[4]).
The DNA thing is interesting but it's more of a "well, this is awkward" footnote to a correction that was already happening, it doesn't seem to be the cause of it.
[0] https://fossilhistory.wordpress.com/2015/01/01/marcellin-bou...
[1] https://www.ebsco.com/research-starters/anatomy-and-physiolo...
[2] https://pubmed.ncbi.nlm.nih.gov/34350666/
[3] https://www.sciencedirect.com/science/article/pii/S295047592...
[4] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6541251/