As the genders of people one deals with become less significant in most aspects of everyday life (discriminating between male and female coworkers could land one in legal trouble, and in the age of remote working, there's the possibility that you might not know whether your accountant or the freelance coder three timezones away you're working with is a man or a woman), this will eventually change, and gender-neutral personal pronouns will arise out of necessity. In English, what will probably happen is that “they” will lose its connotations of plurality, and become the natural way of referring to someone when their gender is irrelevant or unknown.
Not everybody is happy to wait for hypothetical linguistic evolution to take its course; in Sweden, unsurprisingly, they have taken things into their own hands, and introduced a gender-neutral personal pronoun into society, through the child-care system; a generation of Swedish toddlers is growing up used to referring to people as hen (he/she), rather than han (he) or hon (she). The pronoun hen was introduced in two Stockholm nurseries in 2012, and now has spread out of the nursery system to several newspapers; also, it has crossed the border, with reports of it being adopted into Norwegian. (There's a good chance that it'll make it into Danish as well, as it, Swedish and Norwegian are very closely related, and partly mutually intelligible.)
Not everybody is pleased with this, one can imagine the usual conservative talking heads, from Moscow to Wichita, fulminating darkly about “political correctness gone mad”, “Cultural Marxism” and/or “gender” (a term used pejoratively in reactionary circles to mean any deviation from traditional gender roles), in between making disparaging wisecracks involving meatballs and flat-packed furniture. And outside of that, there are some who think that teaching children to refer to people not as men or women but as persons is, for some reason, cruel:
But, argues Dr David Eberhard, a leading Swedish psychiatrist, a new pronoun won’t change the fact that the vast majority of people identify either as men or women. “Whatever you choose to call people, the biological differences between men and women remain,” he notes. “We should treat each other with respect, but ignoring biological gender differences is crazy. Making us identical won’t create more equality.” Boys should be allowed to play with dolls – and girls with cars – if they like to, says Eberhard, who coined the expression “safety addiction” in reference to Sweden’s health and safety system. “But”, he adds, “calling them hen instead of him or her? That’s child cruelty.”
I don't get why this is child cruelty; it's not that a user of a gender-neutral language would not learn to notice that some people are male and some female. The key difference is that this demotes gender from a defining attribute of a person—you are essentially a man or a woman—and turns it into a secondary attribute—you are a person, with a number of attributes (hair colour, height, maleness/femaleness). In a society which is (for the most part) no longer divided into hard-and-fast gender roles, should we still be using language which evolved when the two genders were organised hierarchically, with members of one all but owning members of the other as chattels? That's to say nothing of situations where one does not know the gender of a person (the aforementioned remote coworker), or indeed the rise of non-human personlike entities (with corporate personhood on the books in the US, it seems rude to refer to corporations as “it”, while they are obviously neither a “he” nor a “she”; add to that the prospect of artificial intelligences, which might not always be issued with gendered personae). Finally, one area where a non-gendered personal pronoun would reap immediate, if somewhat trivial, benefits is that of the naming of pets, especially ones hard to sex by superficial inspection (“Nice boa constrictor; what's his name?” “Her name's Ermintrude.”)