Gender neutral does not imply cis-gendered.
If, during conversation, we wanted to specify through a pronoun what gender a person was, there is no way to do it purely with pronouns anymore, since non-cisgendered people will sometimes use "they" so to distinguish between cis and non-cis you need to explicitly state it. Or even to specify the plurality of a group of non-gendered people you have to explicitly state it. Which is not typical english.
If I had two colors on a spectrum, lets say blue and yellow, we wouldn't call the middle of the spectrum "color", we'd call it "green".
> Why would a new set of pronouns be needed to categorize such a non-homogeneous group when neutrality already captures everyone between the normal binary categorizations?
I think this is the issue, it doesn't necessarily capture them directly. I think there is merit in having a pronoun, but it shouldn't be forced by law (as it is where I live). It doesn't arise naturally probably because there are many different categorizations in the middle and most people are on the poles, which I think is fine.