I think you can imagine the tact I'm taking on this. TO cut to the chase, I agree with your counter arguments. I might have a different way of saying them, so here that is.

The person you are quoting is explaining gender identity, that they start off talking about gender in general, throwing a bone to the uses of the term to describe a social construct (of sex-role expectations it seems and not much else), is what I'd call 'the ball fake'. A better way to describe it is they are muddying the waters and creating added complexity to hide the BS they're about to dive into, citations or not. They then get to their goal, conflating "gender" with gender identity, pretending it is a kind of short hand for this property of a person. This matters, because it is a kind of deception (or an additional one on top of the idea that gender has some sort of useful meaning outside of classrooms for young adults as afraid of math as they are of manual labor, and that somehow this is separate from the easier to understand sex role expectations).

So, they are saying this term refers to a socially constructed way to categorize people or their traits, but then say it is also used as a shorthand for some property of a person. Could we say that people have internalized these socially constructed categories? If we did say that, it would invalidate the idea that this is somehow innate or that it is really a trait of the person and not imposed on them or accepted by them from the outside. So, we have skip past that very strong probability, and just accept this notion that gender is a property of a person.

Is this property of "gender" which is shorthand for "gender identity" a separate property from sex? In this person's version, apparently not. It appears to rely on how well it fits with one's sex, because gender is a property "which describes what sexual anatomy the brain expects, whether thats male, female, a mixture of both or neither". Uhm, what? So, the brain has expectations? This person wants to personify the brain as a being separate from a full person. The brain appears to be a manager of some sort hired to manage a body, and at the job interview the brain is told what kind of body it will be governing. The brain responds to the interviewer (God? the body? The body then sometimes lies to the applicant? just ... what the hell is going on here?) saying "Oh yes, I expected that kind of body and as you can see from my gender, I am qualified to manage that kind of body". Then once in the job, the brain says, "whoa, hold on here, this is not job I was hired for! My gender does not match this body, oh the dysphoria!"

Notice also, the person making this personification of a brain is putting these kinds of anatomies as possible bodies this brain's office and desk and view of the city will be in: male, female, a mixture of both, neither. Need any of us address this, beyond simply saying "no, those last two are not a thing and certainly not worth considering"?

We only get here if we accept the notion that it is a property of a person and not a method of categorization created by society. Once here, we're lost and we can never navigate ourselves back to the real world because they keep creating layers of epicycles meant to keep us in this mental prison. By here I mean trying to reason by means of their lack of reason, by accepting the notions we should reject.

If this "gender" is a category for a kind of property of a person, and if it can conflict with the person's property that is categorized as "sex", how do we classify that incongruity? It is not an incongruity that happens in everyone and, without the term being a judgement, it does not happen to normal people. So it is abnormal. To accept these notions, we have to accept that the incongruity is abnormal, and the idea that one's gender and one's sex are in conflict would be the only way we'd know this is a separate category. Every aspect of this seems made up-- to put it kindly. We do not need the epicycles, we do not need "gender" as a property of a person. We never have to accept what society tells us about ourselves, ever.

To put it another way: fuck these fucking freaks, the My Little Pony they rode in on, and there's no need to listen to their bs. It's winter in the north, shovel snow, put some mink oil on your boots or what ever they might need, live a real life as a human with A body, not as a brain hired into a body under false pretense (which is just silly).

I guess the only thing I'm adding to your analysis of that silliness is that: they have to personify a brain, create a category for this property that is not sex but is somehow potentially in conflict with sex, somehow explain how one could know of this category (or what this category says about a person) unless it is in conflict with sex, and some how explain how this category is of the person and not internalized from society, and how it is not a mental illness. So, I picture them as something analogous to a drunken out of shape fool trying to get over track hurdles that are just dumped onto a track at night during a new moon with no lights while trying to keep their momosa from spilling. It ain't happening, and I don't think The Three Stooges would survive such an exercise.

Edited for typos, clarity, but now also to add this: this category of gender they claim for a set of properties might be applicable to a brain and not a person. Otherwise, why talk about a brain as if it was a person?