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I actually really like this idea. Why should gender identity get it's own class of pronouns when no other kind of identity does?
https://en.wikipedia.org/wiki/List_of_languages_by_type_of_grammatical_genders …
I think your definition of "happenstance" must be very different from mine. 
(Totally a guess): Perhaps in the past gender mattered a lot more than other factors did in common day to day interactions.
Or it might be that other factors would have required more than 2 or 3 pronouns -- or didn't apply to as many people -- so were less likely to catch on.
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