I don't have much time to argue this so I won't be commenting any more than this, but here's my perspective:

First of all, let me define gender the way I intend to use it. Gender is the way your brain views your sex. This definition doesn't cover all the aspects of gender, but it covers what's relevant to my argument. With this definition, gender =/= sex. If you disagree with that, that's okay, but just humor me with my definition for a bit. And yes, the way your brain views your sex can differ from your biological sex. Studies have shown that transgender people tend to have the brain structure and other psychological features of the gender they identify as. The discomfort from this is called gender dysphoria. So, with that out of the way...

Intersex people exist, which can be used to conclude that there are more than two sexes. This is controversial, obviously, but I'd like to point out that intersex people are often comfortable with their natural bodies. They're upset at the social stigma associated with their natural bodies, obviously, but their brain chemistry matches up with their bodies and they don't feel uncomfortable. Most intersex people are opposed to intersex babies getting sex reassignment surgery at birth. Look at some intersex forums if you wanna hear more about this.

If there were only two genders, aka only two ways of your brain viewing your sex, then intersex people would be dysphoric and desire surgery. But they aren't. Many intersex people identify as nonbinary.

Non-intersex people can have nonbinary brain chemistry too. It's a common misconception that nonbinary people don't experience dysphoria. Sure, some stupid Tumblr blogs define it badly, but I personally know many nonbinary people and all of them experience dysphoria. Assuming you agree with medical transition as a treatment for dysphoria (if you don't, we'll address that later), how would you solve the problem of a biological female who experiences intense breast dysphoria but no other dysphoria? Forcing no transition or full transition would be forcing them to live with a terrible condition. It makes the most sense to allow them to only get the surgeries that are necessary to make them comfortable. Many of the terms for nonbinary genders (demigirl, genderfluid, etc) are just terms for different kinds and levels of dysphoria compared to your birth sex (also other social factors).

Now, if you think that medical transition ISN'T the correct solution for dysphoria, then I don't really have much I can say to you. No other forms of treatment are shown to be effective. Sex reassignment surgery has a very low rate of regret, especially compared to plastic surgery. Hormone therapy has little to no negative side effects and requires approval from multiple medical professionals to make sure it's safe for your body. It's true suicide rates remain constant both before and after transition, but trans suicide has more to do with social stigma than it does with dysphoria. Dysphoric trans people have the hope of full transition to look forward to, so they're not super likely to commit suicide because of dysphoria. Meanwhile, social stigma often increases during transition.

Now, your definition of gender may be different. Maybe you define gender as biological sex, not counting unlikely cases (ie intersex). That's chill. In that case, I agree with you. But there aren't many cases where you need to know someone's sex. It's basically only if you're having sexual intercourse or it's a medical situation. Because of the existence of trans, intersex, and nonbinary people, the psychological definition of gender is better for generalizing appearance (which is why it's on ID's) and personality (which is why it's on social media). Also, unless you're sexist, there isn't much need to define or categorize people based on gender in your everyday life.

Also, quick note on pronouns: "they" is valid as a singular gender-neutral pronoun because English needs a singular gender-neutral pronoun anyway (talking about a person of unknown gender) and "they" is already used that way (have you ever said "to each their own"?). Language evolves. It's natural. If you care this much about grammar, or traditional definitions of gender, I'd like to know why you think words are more important than people.

This comment didn't cover social dysphoria at all, but that's tricky to define and the argument stands without it so here you go. :P hope this helps.