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> I'm surprised that anyone would really think otherwise

Usually they disprove a much stronger, extreme claim (e.g. some sort of racial purity, or the idea of entirely discontinuous, clearly-delimited races), then imply this means peoples are entirely interchangeable, and that the weaker, obviously-true claim of closer kinship within countries/regions is somehow false and/or meaningless.

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Yep, it is a motte-and-bailey argument[1]. The motte is that the popular notion of race does not map completely accurately onto the spectrum of actual human genetic diversity, which of course is true. The bailey is that the idea of race is completely meaningless and that anyone who buys into it to any degree whatsoever is a racist. The bailey is often presented using the thought-terminating cliche of "race is an unscientific/pseudoscientific concept".

[1] https://en.wikipedia.org/wiki/Motte-and-bailey_fallacy

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