Told reading tests are a game, boys score higher

This is a very broad statement which is trivial to prove false. Look at powerlifting: men perform better than women, and testosterone plays a well-understood role in this performance. This is a function of sex, not gender. And while it's most numerically provable and obvious in powerlifting, this is true for a lot of sports.

At a more fundamental level, I question whether the gender/sex divide is scientifically valid. One can point to the example of a transman and say, "Their female sex didn't align with their male gender at birth" but that seems like a very surface assessment; it's also possible that chemicals released by (for example) their brain or pituitary, are causing the dysphoria they experienced. Are we to claim that ovaries and testicles are sex, and the brain and pituitary is gender? Or might it make more sense to broaden our understanding of sex to realize that genitalia aren't the only organs regulating sex, and instead say that the transperson wasn't born female, but was instead born with a mixture of sex characteristics, and only the most obvious sex characteristics were female, while the characteristics which more fundamentally defined their identity were male.

In a larger sense, this seems related to the categorization error of separating mind and body as if they are two different things, a separation which seems more clearly false.

NOTE: Please don't construe anything I say as negative toward transpeople. I think humans have an inherent, fundamental right to define their identities and modify their bodies in any way they choose. People don't need excuses or justification to exercise that right. Certainly we don't need an abstract distinction between sex and gender to let people dress and socialize with people the way they want or change their bodies.

   intersex is considered a very normal

Sexual reproduction so as to perpetuate humanity is the single most important function a society executes. All human culture directly or indirectly relates to this task. One of the key tasks in this context is the choice of partner(s) whom to reproduce with. Alas, not any two humans can reproduce together. Indeed, an exact determination who can successfully reproduce with whom is still not possible in 2016, although we are much better at this task than say 1000 years ago.

Given that we can't predict mate compatibility perfectly, humanity needs heuristics to make this decision efficiently. The following heuristics are being used in practise:

- Divide humans into two, male and female, based on the shape of their genitalia.

- Have sex with members of the opposite group.

Humans use additional heuristics for deciding with whom to mate, e.g. social status, age, but the male/female binary is by a large margin the most important.

As you correctly point out, not all humans neatly fall into the genital-shape based classification between. But it doesn't matter, because the heuristic works so well that humans can reproduce successfully by ignoring those outliers. That means outliers must be infrequent enough (say > 99% fit the binary).

I can't overemphasise this: the heuristic has been extremely successful.

The very concept intersex makes sense only in contradistinction from binary normalcy. To see this, consider the fact that most humans have 10 fingers, but not all. Yet, there is no socially relevant concept of 'interhanded' people, no pressure groups, nobody claims to have been born with the wrong number of fingers, no finger reassignment surgery etc. That's because the number of fingers is not all that important.

In the future we might have a refined classification, as our understanding of human reproduction has improved immeasurably in the last century. But note that a finer classification still relates to biological facts about human bodies. It would probably be something along the lines of

- Can give birth to children, i.e. falls in the category woman, despite non-standard genitalia.

- Can father children, i.e. falls in the category man, despite non-standard genitalia.

- Can neither father children, nor give birth to children.

In other words, it would be a refinement of the male/female binary.

What evidence do you have that identity doesn't originate in biology?

I'm well-aware of the existence of transpeople, but I'm not ready to conclude that gender dysphoria doesn't originate in biology. There's more to a person's biology than their genitalia.

"Hey Teacher, how much is the curve on this test gonna be?"

"Well, it won't be anything because John got a 100."

[Angry glares at John]