And I'm being picky about how you're picking on them picking on language.
Your complaint boils down to wanting a "fair face-off between two terms". That already grants that the two deserve a fair face-off. Not all pairs of things deserve a fair face-off. Let's have a fair face-off between 5G causes Covid and viruses do! Let's have a fair face-off between young-earth creationism and all of science! Let's have a fair face-off between campfires and ubiquitous rooftop solar panels!
No, let's not, because implicit in the idea of a fair face-off is that the contest is meaningful and close enough to be in doubt. The authors' contention was that it wasn't close, that "sex assigned at birth" was a stupid ideologically-driven term, so it's entirely reasonable that they used scare quotes. Indeed, this is good style: it telegraphs their perspective that this term, alone, is the one deserving of scrutiny. If they had scare-quoted other things, it would have telegraphed that they intended to doubt both, or do a head-to-head test of both, which was clearly not their intent.
So, from a matter of clarity and good writing style, they made a sensible choice. But they might have made a factual error, because maybe there is good reason to think the two are an roughly equal footing, so their targeting of one was misleading: good style to telegraph their intent, but their intent was bad because their perspective is counterfactual.
Therefore, the only sensible reason for you to criticize them is if you believe that it is a fair enough contest so it ought to be a head-to-head, which would involve one or both of (1) a vast overestimation of the utility and accuracy of "sex assigned", or (2) a substantial underestimation of the utility and accuracy of "biological sex".
Given that the authors' complaint is that the people who like "sex assigned" are trying to cast doubt on the reality of "biological sex", and given that the only thing you said about "biological sex" was a reference to a completely specious attack on it, the natural inference is that, indeed, you very possibly were trying to cast doubt on its reality. You don't need to flat-out say so in order to present a strong inference. It's true, you didn't say it doesn't exist. But you implied it is a dodgy concept.
I'm not sure you got my point about "assigned as rainy". There are, of course, some cases where the dichotomous label for a smoothly varying phenomenon are going to be ambiguous. And then, yes, "assigned as rainy" is a perfectly reasonable way to talk about just those ambiguous cases. But if you turn on the weather report, you expect to hear, "tomorrow will be another rainy day," not, "tomorrow looks like it will be assigned as rainy, just like today". If it's going to be close, you expect entirely different language used to get the extra precision needed, e.g. "expect a few sprinkles tomorrow". To use a non-weather analogy, if someone named Patterwell got 62% of the vote you expect to hear "Paterwell won the election" not "Patterwell was assigned as the winner of the election". The latter language is used to highlight the unusual arbitrary case (e.g. "George W. Bush was assigned as winner of the 2000 election.").
My point is not that "sex assigned at birth" is never the relevant concept. Only 99.98% of the time or so, it's not the relevant concept (as I explained last time). The 0.02% where it is--sure! Use it! The point is that it is the ubiquitously used concept now.
Maybe it's mostly in liberal areas in the U.S., California and Colorado (Denver) and New York and such, where people do things like write user interface guidelines and medical form language guidelines, and you live somewhere else?
And I am, maybe not angry, but rather annoyed by this, because the change was a result of a lot of pressure by advocates (bottom-up, not top-down, but definitely pushed by collective activism making people feel that going along is the compassionate thing to do, not an organic "everyone understands this is the sensible thing to do"--yes the authors did an awful job of documenting this but I've seen multiple instances first-hand), and all that effort could have been used to get something helpful instead of something that is almost always misleading and almost never medically useful.
If the pressure had been motivated by medical relevance, there would have been a lot more thought given to, for example, conveying the status of gender-affirming care, which acts biologically, not culturally. It impacts breast cancer risk, probably heart disease, likely influences metabolism of drugs, etc., and if we had better information on status we would have a way easier time discovering other important factors that might be relevant for well-being instead of having to convince the NIH to fund a speculative study.
A high-pressure effort to enact a policy that pushes, by implication and scope of deployment, a counterfactual view of reality and fails to add needed distinctions to help get people good medical care is, absolutely, the kind of thing that bothers me, and if people dig in their heels rather than try to come up with something well-aligned with reality and with medical relevance, that bothers me too.