Gender neutral pronouns are a necessity that we have too long ignored. My understanding is that most languages have these. I can't tell you how fucking sick and tired I am of writing "he or she" in legal documents.
You could simply switch to Finnish. There is no gender differentiation of the third person pronoun in Finnish.
The sub dialect of the village I grew up in has only the equivalent for he and his. Works for both males and females. There are no words for she and her.
Funny, I had noticed it as a kid, but was raised in a little different dialect. Thought it was weird people had no idea when to use her and she. Found out after I left home that it was a particular thing of the dialect in that small area.