I've written over 300 articles on the anti-trans agenda. From memory, here are three that first come to my mind as potentially related to what Byrne and Hooven are doing in rejecting the term "assigned sex." (These are unpaywalled "friend links.")
‘Gender-critical’ ideology is body/mind dualism
Peter Boghossian treating the word "trans" as a synonym for "fake":
And maybe also this one...
‘Cis’ or ‘Biological’?
Given that extra context, and to keep it brief, I suspect Byrne and Hooven are saying that "sex assigned at birth" is a fake concept — and by extension (given our political context), they fully expect others to leverage this small language recommendation to make a broader suggestion that being transgender means having a fake sex/gender. If "trans people are fake" is someone's starting belief, that person will tend to preclude all vocabulary whatsoever that's meant to describe "trans," since, in their view, nothing "trans" ever exists and thus can't be properly described — unless those words denote "fake."
Some trans people are intersex, or believe they might be, or (consciously or not) draw from cultural narratives of trans-ness that originated in early 20th-century beliefs that (incorrectly) equated being "genuinely" or "legitimately" trans with being intersex. Also, certain physical sexual traits (if not necessarily one's "sex" comprehended as a whole) can indeed change. Hormone levels can change, for example. "Sex assigned at birth" is a phrase that can be used to neatly communicate this complexity. Without it, one has to say things like "I'm female because I was born female (end of story)" and then, despite relevant ways in which that same person may also be validly understood as male, their apparent maleness will be more likely to be assumed to be wrong and fake, so the ways in which they live as male (and/or as a man) will be socially delegitimized. That is what will happen if we throw an entire term into the trash bin without understanding or caring why some people use the term.
