After 32 generations (~800 years) you have more genealogical ancestors than there are base pairs in human DNA. There literally isn’t enough resolution to store a “record” of each of your ancestors, even if you inherited exactly 1 base pair from each ancestor.
Additional complications make it an even shorter timeline. After about 8 generations, you share no more DNA with your ancestors than you do with a random stranger.
Politically this should be more well known. Of course, racists and fascists rely on “blood quantum” arguments to justify racial or ethnic oppression. But they don’t invent this idea of strict genetic identity. It’s latent in the population.
Leftists should more frequently call out genetic tests like 23andMe as inherently racist because it’s based in race science nonsense. It may not be as obvious as Nazis invoking aryan genes or whatever; but it’s still just as incorrect when your aunt at Thanksgiving talks about how she discovered she’s 2% Choctaw or whatever
Dawkins is a piece of shit for a number of reasons, but where did he ever say anything like that?
Perhaps I'm misremembering, but I'm pretty sure he describes everything about life as just a means for DNA to propagate itself. As if DNA isn't merely a component in a larger, more complex and complicated system, but as the reason for the whole thing.
You're not misremembering because those were major misconceptions about what he was saying, especially in the premise of the book that made him famous, The Selfish Gene. But that book as well as his other ones on evolution, genetics, and biology are accurate, with Selfish Gene just being a popular-accessible explanation of the scientific consensus on the role of genes in evolutionary biology. It was misconstrued, often intentionally and said to be this nihilistic justification for human selfishness, which it's not, and I think that's what you're remembering. It's one of those instances where someone who may be an asshole and totally worthy of criticism gets criticized for mostly the wrong reasons, and those wrong reasons are what end up getting solidified in memory incorrectly as why they're an asshole.
Replicating molecules (like DNA) replicating and propagating themselves is the "reason" for the whole thing in terms of how biology at the molecular level works. It's not an answer to a "why" question, but a description of what's really going on at scales we aren't used to thinking about or readily able to see. But neither Dawkins or any other evolutionary biologist I'm aware of ever implied that DNA is all that "matters" or that or that other systems aren't at play in reality. Just like the fact that "all the matter we see, including life, is composed of interacting atoms" does not negate the presence of society or human emotion, neither does the genetic description for the foundation of life and how it evolves.
But all of that is aside from Dawkins' Islamophobia, misogyny, deep misunderstanding of systemic racism (i.e., just racism), his flippant disregard of (CW) SA victims with the justification that he was one himself and therefore had the right to extrapolate everyone else, his creepy views on human sexuality, etc. In other words, Dawkins is a shit person and absolutely should be called on all the disgusting nonsense he's said, but let's call him on what he actually got wrong and not perpetuate misconceptions about all the stuff he got totally right regarding evolution and biology, which ultimately just carries water for the chuds and creationists.