Phew, I caught this just before posting a literal essay. I do hope @SillyStrings423 comes over as the debate has actually ended up a lot more promising than it started out.
@SillyStrings423
Then, as the child grows up, they may realize they feel uncomfortable with the gender identity they had been assigned based on their genitals, because it is at odds with what they know they'd feel more comfortable with. That feeling persists in transgender people, no different than repressed homosexual feelings persist in gay people.
So there's a few things to unpack here.
There is the body's physical sex - fixed at conception and binary in humans (in rare cases the physical expression is unusual due to a DSD, but every human is either male or female). (A)
There are the stereotypes and expectations our society attaches to the physical sex of the body - what you call the "social categories" and Gender Critical people call "Gender". (B)
There is the Gender Identity, which Genderists believe to be an innate sense of - something - that makes one akin to others of the same gender and different to those who are not. (I'm not a genderist so I may have this wrong. Please feel free to clarify). (C)
I agree with you in today's society, historic sexism means that Man and Woman are social categories (B) as well as simple body types. (A)
If I understand properly, both Genderists (those who believe in an innate gender identity) and Gender Critical people (those who believe Gender is always a social construct) think it's wrong that A and B are linked together. Both agree that while your body sex (A) exists, it should not force you into a particular social category/gender (B).
So far, so the same. The difference is in how each group thinks this should be resolved.
Gender critical people either do not believe that (C) exists, or believe it exists but is a personal thing that should not be a basis for laws that overrule sex. (Please be very clear here that believing (C) does not exist is not the same as believing trans people themselves do not exist! As an analogy, I believe nuns exist and are genuine people whose faith is honestly held and makes a real difference in the world, but I do not share their beliefs about the source and purpose of their, or my, existence.)
So the gender critical position simply is that we should get rid of (B), leaving Man and Woman as names for certain body types, (A), with no social element. For GC people, (A) is fixed and so has to be dealt with, (B) is a social construct that only serves to limit people and we would all be better off and freer without it, and (C) is simply an individual aspect of someone's personality.
From that perspective, a trans identity becomes unimportant not because gender is fixed, but because there is no difference between men and women other than those which are consequences of the bodies we have. The trans person was always valid and does not have to change anything to be seen as who they are.
However, because male and female bodies have different capabilities, and because female-bodied people suffer from society's sexist history, Gender Critical people also accept that even if we reduce to (A) there will still be times we need to retain sex-specific rights or protections. The important thing to understand is that from a GC perspective, when we distinguish between men and women it is nothing to do with any individual's personality or self-image, just a practical distinction based on the sex of the body for times when the sex of the body matters.
This is a longstanding Feminist view that predates the current trans movement by many decades.
The Genderist view is, I think, that we should keep (B) and (C) but get rid of (A). Everyone becomes free to choose a social sex category (B) that suits them best based on their gender identity (C). For Genderists, body sex (A) is irrelevant because they do not believe it gives rise to significant social or physical consequences and therefore it can be ignored outside some medical settings. Female people do not need any social, political and legal protections.
However gender identity (C) does have significant consequences and mean for some things humans need to be separated by gender. (I'm not sure why, nor why those "things" happen to be exactly the same things that were historically based on sex. Would be great to have that explained.)
So the clash is not about one side wanting to keep stereotypes based on sex and the other wanting to lose them. Both sides want to lose them. The clash is about the best way to do that.
So, to get back to our uncomfortable child...
Genderists would say their discomfort is because the child was "assigned" the wrong gender (B) based on an incorrect link between sex and gender (A) and (B). To be fully comfortable the child needs to assert their true gender (C). (For some reason this also often involves changing cosmetic aspects of physical sex, which I don't understand since it's the assigned gender that is supposedly wrong rather than the body sex - I wish someone would explain that...)
Gender critical people (those who do not believe in gender identity) think it's because the child is reacting against the social limitations imposed on them through sex stereotypes (again B, but this time from a gender-critical perspective). To be fully comfortable, the child needs society to ditch the social category aspect of sex (B) completely.
I can see why Genderism is attractive. It's the easy option. An individual only needs to worry about their own transition, not the wider picture. It gives the illusion of progressiveness and acceptance without having to force society to ever properly break down those social categories. As feminists can tell you we've been trying to smash them for decades and the beggars are tough!
But it's the Gender Critical position that is genuinely progressive.