How about this argument?
I think you're just pushing the social construct down (up?) a level from phenotype to genotype. The fact that phenotypes are reflective of genotypes is a trivial observation. The fact that genotypes are geographically distributed is a trivial observation.
The fact that a particular constellation of phenotypic/genetic characteristics get lumped together and called 'race' is a social construct. Granted, the phenotypic variations that we call 'race' are generally pretty glaringly obvious, (as opposed to say, innie vs outie belly-button), but that doesn't make it any less a social construct. Not a particularly useful one, either.Interesting point. A few responses.
I think that most people, if they bothered to give serious consideration to the question, would readily agree that phenotypes are caused by genotypes (e.g. dark skin vs. white skin is caused by genes, not just magic or sun exposure or nutrition) and that phenotypes have geographical distribution (i.e. there are more dark skinned people in Africa than in Iceland.)
I think that if you pushed the point with them they would probably also be forced to conclude that these two premises indeed imply that certain genotypes must also have geographical properties (whatever genes cause dark skin are more common in African countries than in Iceland). Add in the assumption that geography is related to ancestry, and that one way of thinking about race is as a crude description of where most of your ancestors lived 500 or so years ago, and we're a long way to a good understanding of the issue.
I would assert, however, that many people do not actually seem to display such understanding in the way they discuss the matter, notwithstanding that you could convince them of the truth of each premise. When you point out the conclusion, they still act surprised. Acknowledging that C follows from A plus B is different from people instinctively believing C. Even if race as popularly described were nothing but skin color, as long as that's genetic, would you really describe conclusion C as being consistent with 'race doesn't exist' or 'race has no biological basis' or 'race has no genetic component'? It seems like a bit of a weird stretch.
And the reason this seems striking to me is that I've actually had conversations with quite intelligent sociologists who started out the conversation asserting that race didn't exist, or that the fact that there is more genetic variation within each race than between them meant that race was meaningless. When I posed the conundrum below, they appeared to have genuinely never considered the paradox. They were truly puzzled, and didn't have any answer.
I don't mean to be trite, but nothing in your argument actually answers the narrowly defined question. 23andme is able to reconstruct, to a high degree of accuracy, analogous descriptions to the ones people use such as 'black', 'white' and 'asian', out of purely genetic information. I never asserted that race is not partly a social construct. It is. But that is very different from saying that race is purely a social construct.
Race as popularly described may focus more on some phenotypical variations than others (as you note with skin versus belly buttons). But people still seem to manage to identify most of the main principle components of genetic variation in the labels they attach. In other words, even if 'race', in terms of how people describe it in common speech, is just a crude description of how you look like, that description seems to be correlated with the various principle components of genetic variation. That's the key part. If 23andme had merely identified the genes for skin color, then attaching race labels that correspond to skin color would be a trivial observation. But my understanding is that they don't look for these specific things, but large clusters of genetic variation. That's why they're able to say much more about the full breakdown of your ancestry, rather than just 'your skin is probably brown-ish'.
In other words, the labels that people attach are indeed correlated with large principle components of genetic variation, which are in turn associated with self-reported descriptions of ancestry. Which is exactly what you'd expect if those genes were associated with groups of people who had been geographically separated for extended periods of time. Which, of course, they had been.
From this point of view, the real information is of course in the genes, not the crude description. In other words, it's much more useful to identify the genetic information if you want to say meaningful things about someone's likely characteristics, rather than just the socially defined markers of appearance. Once I know someone's full genetic information, there's not informational content left in the popularly described concept of 'race' (other than than purely social effects like cultural traits). But that doesn't mean that the socially defined markers are worthless if you don't actually have the ancestry or genetic information.
Seen this way, the only real remaining question (and it is a large and separate issue) is the usefulness of these classifications. If you buy the argument that these classifications are picking up large principle components of genetic variation, do you really think that such variation would have no useful predictive power at all? It's possible, but it only would seem likely if you think that genetic variation itself don't matter much - that it's all environment, in other words. That's a whole separate debate, and entirely possible, but my reading of the literature is that heritability estimates of around 50% for lots of characteristics seem to suggest that it's not entirely environment. Even if it were, though, I still get to my initial conclusion - what people identify as 'race' is indeed partly genetic, because it's highly correlated with genetic variation.