Singular "they" has been used for hundreds of years dating back to the 14th century -- comparatively, it's a very modern trend for people to say that it should only be used in the plural. Even forms like "themself" have a pretty long historical precedent.
> what purpose would a "gender-neutral" pronoun serve?
From a technical perspective, these pronouns allow you to refer to a single person without prescribing their[0] gender. This may have the effect of making some technical writing more accessible.
From a narrative perspective, these pronouns allow you to write about a character who's gender you might not want to reveal yet[1].
And from a political perspective, these pronouns allow individual to assert that they are nonbinary. The political perspective is what gets people annoyed, but it's worth noting that both language and politics are a reflection of society. It's normal for one to influence the other. It's because of this that we consider language and grammar rules to be descriptive, not prescriptive -- they simply describe the words that people use, even if those words are being used for a political purpose.
Ignoring politics, why not use "he" as the gender neutral form and get rid of "they"? For the same reason why it can occasionally be annoying for `0` to be automatically cast to the boolean false in languages like Javascript. If I use the word "he", I don't want someone to have guess whether I'm using it in the gender neutral form or if I'm literally describing the person's gender. Singular "they" serves as a strict typecheck -- it makes "he" into a more useful word because you can use it exclusively to refer to male subjects.
[0]: Also note the usage of singular "their" here, which is almost universally accepted. I've never heard someone argue that we should get rid of singular "their" just because we already have "his" and "her".
[1]: "The cloaked thief stepped into the house. Their eyes scanned left and right, as if they couldn't decide what they wanted to take first." Is the thief a man or a woman? Only the detective will know!