Robert, the argument to which you allude is the classic divide between “race” and “ethnicity.” And as counterintuitive as it sounds, it is not wrong to say that one’s DNA is “not integral to race.”
I could show you two people from the same ethnic group whose skin color is quite different, even while their coincidence of DNA is greater than that of one of them and a person who looks more similar racially, but is from another ethnic group.
For instance, removed from their geographic context and thrust into a true “melting pot” of cultures, one might closely identify, on a racial basis, a Pashtun with a Castillian Spaniard from Malaga, more than would that Spaniard be racially identified with his countryman from Santander.
And as nauseating as the term “social construction” is, I think it is beyond question that the importance of one’s race, insofar as it is important at all, really is largely determined by the society in which one lives. This, as I’ve understood it, is the crux of the argument when one claims that “race is a social construction” — it does not literally mean the color of your skin, but how that color is socially perceived. Black in Africa does not mean the same as black in the United States, as any African living in the U.S. will tell you.