Do you also want to encode race, country of origin, height, eye colour, birthdate, into every sentence? Why is gender special enough that it’s worth the problems it causes for many?

Gendered pronouns cut the search space in half which makes them an efficient way to avoid ambiguity. Other traits aren't so clearly split. Gender is distributed roughly 50/50 across the entire population, regardless of nationality, skin color, eye color, etc.

You could phrase the question this way if @trianglem was suggesting to make some language more gendered. It totally makes sense to ask, why gender is special, in such situation. But what they did is different, they asked why do people want to make a language less gendered. Which is a completely valid question to people who advocate this.

Some languages already do make distinction between genders, and this is a feature that makes gender special enough, unlike height or eye color. Making changes to a language requires effort and causes inconvenience, so it's up to advocates of making a language gender-neutral to provide a motivation which would be worth the effort. If there was a language which had different pronouns for people of different height or eye color and we wanted to change that, we would have to motivate this too.

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I personally think that it's convenient to have both neutral 3rd person pronouns, which can be applied to anyone, and pronouns that make distinction based on something (not necessarily gender) as well. Neutral pronouns are useful in cases, when you don't know enough to choose the right non-neutral pronoun. Non-neutral ones are good because they give you a convenient way to make distinction between things you talk about, as @theli0nheart already noted here.

For example, in English you can introduce two people of different gender and then refer to them as "he" and "she" without having to repeat their (possibly long) names. In Turkish, where they have only one singular 3rd person pronoun "o", this won't work and you'll have to either name the people every time or use some other shortcuts. In Russian, for example, you can go even further than English — in this language even nouns, which do not denote people, have gender, so you can conveniently refer even to objects in your speech.

As someone who occasionally reads but usually does their very best to avoid this topic due to its tendency to be a total minefield, I just wanted to let you know that this particular argument doesn't work for me at all.

You'd be better acknowledging the fairly obvious benefits that come from encoding so much more information into a single word, and then going on to argue why despite that it would be a good idea to move towards neutral suffixes.

When speaking Spanish, for example, I definitely feel the lack of information conveyed by su when compared to his/her.

It's not obvious to me that knowing everyone's gender at all times is important. In fact, as you say, there's a variety of languages where that isn't the case.

When I'm talking about someone I bumped into on the bus, why is it relevant what gender they are above pretty much anything else you could choose to encode there? I'd much prefer to encode, for example, my relationship with them - whether I'm close to them or not, whether I see them as equal, inferior or superior.

Gender is mostly a thing that becomes relevant when talking about relationships and sex or healthcare, and even then not always.