We used to taught that race was constructed. There is no biological basis for race, except the fact that certain populations of the human race share genes for certain traits that make them look different. But there is about as much genetic variability in Africa as in the rest of the world combined. Racial categories are the product of racism itself, as we can see when we call someone black who has a black father and white mother, and lump in East Africans with West, as though they were the same "race" because of a few visible characteristics that are superficially similar. Other racial categories are deeply confused with religion, language, and cultural identity. For example, someone who is "Hispanic" comes from a country where Spanish is spoken, but could be of any ancestry group or any combination of them. I have friends who are American Indian, but with a vast majority of their ancestry being not American Indian.
We were taught that sex was biological. Once again, human, like other mammals, come in two sexes. Nobody seriously argues that there is not a difference between a cow and a bull, and that this difference is not biological. Why is this different for human animals.
We were taught that gender, the set of social and cultural conventions we use to organize this sexual division, is not biological. For example, gender in language is a classificatory system, that is optional since not all languages have it. Wearing a particular kind of shoes is a gendered thing, because when you think about it men's and women's feet are pretty similar to one another, so why are there women's shoes?
With Judith Butler, we get the idea that sex, too, is a construct, so that the constructedness of gender goes all the way to the bottom. To demonstrate this, she used examples of people who don't fit the male / female divide very well. (Simplifying her thought greatly here.) This radical anti-essentialism, with the accompanying idea that gender is performance, turned into a very essentializing tendency. Not only can any individual perform any gender it wants, but can also transform itself biologically into the other sex. And if biology has nothing to do with this, then taking hormones or having surgery should not enter either.
So when Tuvel asks why race cannot be performed in the same way as gender, I have to agree with her. Don't the same arguments apply? Why does gender get to be an individual choice, and race not? I guess it's because the two communities of scholars she was accused of ignoring have decided to frame their ideas very differently, making impossible any comparison between paradigms. It would have been smarter to write a philosophical article on the incommensurability of two essentialist paradigms, but that doesn't justify her treatment.
You can still see race as an arbitrary construct created by racism, as it is, and argue that people who share the historical fate of having had their ancestors' race defined by racism still get to use racial identity to further their aims. You can see sex as biological and still advocate for people to perform whatever identities they want to, and to make biological adjustments to make this performance easier, or to change their biology itself. You can even argue that someone can change their racial identity if they want, and simply debate the issue, with opinions on either side, letting the better arguments have the day.