Main problem in your argument is the insistence that sex and gender are discrete categories, one biological and one socio-cultural. The problem with that is biology doesn't exist in a vacuum, and human biology doesn't exist in a socio-cultural vacuum. Rather, society and culture impact biology such that there can be no pure "sex" that isn't already gendered by society and culture. The engendering of sex starts prior to conception, contributes to the development of the fetus in utero, and continues to shape biology, anatomy, and physiology after birth through til death. Gendered regimes of diet, exercise, nutrition, and violence ensure that the sexed body is always at least partially a product of socio-cultural gender. Pretending the two — sex and gender — operate independently simply is not supported by scientific evidence or experience.
That doesn't mean biology is infinitely malleable by society or culture, just that treating sex and gender as unrelated aspects of the human experience ignores what research has shown is a dynamic, interactive relationship between the two. What queer theory contributes to this conversation is its insistence on understanding nature as a social construct, not some pure, unsullied terrain of fact or truth that precedes society or culture. In fact, there is no nature without culture, just as there can be no sex without gender. The second is what constructs the first. Oh sure, there's a material reality 'out there' that we've decided to call "biology" or "nature" or "sex." Or, at least, that’s what our sensorium tells us is the case. But the moment we start deciding that some biology is "sex" while other biology is not, gender has entered the chat.
Finally, I think you mis-state the "goal" of queer theory. Queer theory isn't some static, unified field where all agree on a goal or purpose. In fact, to the extent that it's still extant, queer theory is an ongoing, dynamic conversation among scholars who don't always agree and often didn't even know they were working in the same field. But none of them I've read has the goal of abolishing sex. Rather, there seems to be an interest in understanding sex as an inadequate system of classification that does violence to some bodies in the interests of (re)producing the cultural fiction that human biology is binary in form and function. The purpose of this fiction would seem to be the naturalization of societally produced gender differences by anchoring them in some timeless, pre-social nature called “sex.” Revealing binary sex as a cultural fiction doesn't abolish “sex.” It helps us to see that “sex” doesn’t exist outside society or culture.
