To be clear: the word 'assigned' bothers me, too. I agree that our sex designations have always traditionally been a reading of what we see, not something we simply decide for people. That's why I was offering some other terms.

Regarding morphology: Yes, with adults, we usually use morphology, rather than the genitals. But I'd argue that morphology is also ultimately just a heuristic for gametes.

If I was the dictator of language, I would probably say we should base it on gametes, because that's ultimately why sex exists at all, and any other types of morphology or behavior around sex is ultimately to get two types of gametes interacting with each other.

And if we used that definition, then genitals, morphology, and sex / gender identity (or "brain sex), would all just be useful heuristics for predicting the actual sex - i.e. gametes.

But as you said, language is partially made to suit how humans see themselves psychologically and culturally. And that's why I think it's important to include trans people's own psychology in that calculation. And to that end, I don't think it's unreasonable to consider having something like "brain sex identity" be the source of truth on sex in humans - rather than gametes or genitals or morphology - even though those may be more consistent with history and non-human biology.