Indeed, we might not be talking about the same thing.
The problem with Byrne & Hooven's article is:
1. They say that the term "assigned sex" is used in "some medical and camp forms," "academic articles," and medical organizations. They don't quote a single example of such sentence, so we can't hear and assess the language in context. This observation is so broad (a term exists!) that the assertion allows/requires the reader to draw from their own frame of reference as to what's being talked about. Whether I'm personally aware of the various real-life uses of the term is not what I'm placing at issue here. My point is that Byrne & Hooven should have better defined the scope of their complaint. They wrote the NYT column, so they are the ones responsible for knowing/teaching about their subject matter. If they don't take the opportunity to do so, each of us will insert our own frames of references into the gap they left.
2. Without having shown us a single example of how the term "assigned sex" is being used (I mean, they haven't quoted a full sentence and provided context for it), much less given us a comprehensive review to show us that different people might mean different things by it in different contexts, they assume that "sex" is the "language that more faithfully reflects reality." This is a way in which they shift the burden onto their reader to already know what they're talking about, and then to assume that their preferred language reflects reality and other language doesn't.
3. They say that "one reason" for the use of the term "assigned sex" is that it provides "emotional comfort and insulation from offense," particularly for trans people. They simply assert that this is the motivation; they don't support this claim. Having assumed that this is a principal motivation, they go on to assume that it's an inadequate one. Then, they offer a reason not to "promote" the term "assigned sex," which is that non-transgender people might feel "shamed" for not using it. (They're deploying the classic gambit that the minority's feelings don't matter and the majority's do.) Also, it seems they believe that discouraging the term "assigned sex" will have relatively little negative effect on discourse, even as they insist that discouraging the term "sex" supposedly "repress[es] the very vocabulary" for the general population "to discuss the opinions in the first place." (Because trans people's opinions are not "the opinions" worth discussing?) Their request is for "professional organizations [to] change their style guides and glossaries" and for "journalists, medical professionals, academics and others" to similarly use the word "sex," not "assigned sex."
Essentially, they are saying that everyone should always say "sex," and no one should ever say "assigned sex." In their own academic careers, they have considered counterarguments, but they chose not to reflect any of those counterarguments within their opinion column. They present their recommendation that everyone else should entirely jettison a term without examining nor admitting any possible exceptions or nuance for why other human beings use the terms they use.
A core assertion of Byrne and Hooven here is that everyone always worries about trans people's feelings getting hurt and no one is sufficiently worried about non-trans people's feelings getting hurt. And that this misplaced concern (on the wrong people!) leads people to say things that are fake instead of real. And that everyone should stop using improper language and use the language that Byrne and Hooven tell us is proper.
The NYT and other newspapers and magazines routinely publish non-trans people's perceptions of trans people that work roughly like this. Typically, they are poorly argued. This is the latest example.
That's what I'm talking about.