While languages change all the time, they are very resistant to people trying to impose change upon them. Consider l'Académie française, which has been trying unsuccessfully for decades to get the French to stop saying le weekend and similar Anglicisms. While there have been numerous attempts to introduce a third-person gender neutral pronoun, none of them have gained any traction.
Meanwhile, the use of singular they, which had been used in English for centuries, but which had been discouraged by grammarians for around a century, is now undergoing a renaissance, to the point where it has become widely acceptable among young people. See this language-log post.
While using singular they will occasionally give rise to an ambiguity, any attempt to get people to start using any other singular gender-neutral third-person pronoun is tilting at windmills.