Oh yes it does.
An antenna is not a leg--well, unless you're a fly with the Antennopedia mutation, in which case your antennae are legs, and you are pretty much screwed.
Hair follicles are not skin, and the distinction is mantained by a variety of mutually inhibitory feedback loops.
Nature cannot help but "like" cleanly divided categories because some things function well to enable perpetuation of the species, and some do not. That comparatively rare exceptions exist only indicates that biology is a messy inexact process, and "pretty good" is good enough.
Everything is, in a sense, socially constructed, but it does not follow that it's arbitrary. You cannot reason from arbitrary constructs: they exist in contradiction to underlying patterns, and where the borders are is also entirely a matter of convention. In cases where there is already a lot of convention layered on top of some strong underlying pattern, it may make sense to separate the two which we already did by distinguishing gender and sex.
To then go back and decide that, no, we're going to throw away the physical distinctions too because, well, we found an exception and anyway it suits our ideology, is fundamentally anti-rational--and harmful, because it directly impedes our ability to do things like recommend screening for ovarian cancer and breast cancer.
That a doctor "decided" the sex of a child makes it sound like it's that doctor's whimsy. That there are a small number of ambiguous cases does not warrant this characterization: in most cases it's highly obvious, and the doctor, the nurses, the parents, the relatives, and any random passerby who happens to see when you're changing their diaper will all come to exactly the same conclusion. And this is precisely because nature loves having a cleanly divided category here, mistakes are rare, and people make the call of sex on the basis of this highly but not completely reliable division.
The motive to fail to recognize quantitatively those patterns that exist in biology seems to me to come from an ugly, hate-embracing intellectual framework.
Supporting people's rights does not entail pretending that there are no differences between them and others. Indeed, exactly the opposite is true: when we support individuals' rights we accept that there are differences and that this is okay.
If it is okay to be cruel to people if we recognize that they have atypical physical development, or a mismatch between neural and non-neural development, then we are allowed to be cruel to people with all manner of disabilities--or we have to pretend that all manner of disabilities are actually just "normal".
Neither is true, and neither is kind.
We can accept that biology is what it is, label it with straightforward labels, and still recognize that there are rare exceptions that need to be treated with compassion.
And the social constructs layered on top, which are far more arbitrary, can be molded however we desire to attempt to achieve a more just and empowering society.