This is a serious omission; not least because sometimes, you want to talk about someone you haven't met yet, and don't know whether to call them "he" or "she". Or even whether it makes a difference which order you mention the two in. Perhaps I should have said "..... and you don't know whether to call them 'she' or 'he'". And this is before we even begin to consider that even though we might have met the person, they might not want what is between their legs shoved in everyone's face.
Now, there have been various suggestions for sets of common-gender pronouns; especially, it seems, in science fiction. There's no reason why aliens should follow the same binary distinction that worked well enough on Earth. Yet none of them have really caught on — despite there being entire science-fiction anthologies without a "he" or a "she" anywhere in sight.
But hold on a second. The answer is right there, two paragraphs ago, when I referred to the hypothetical unmet person as "they". We do it all the time in informal English. So why can't it just be accepted more formally?
Outside certain dialects, English no longer makes any distinction between singular and plural in the second person, and the distinction is often blurred in the first person (traditionally, the reigning monarch has always referred to themself as "We"; and this is also often used as a literary device to indicate that the author is not alone in expressing a particular opinion. There are also instances when multiple authors have referred to themselves as "I".)
- Nominative: They
- Accusative: Them
- Genitive: Their
- Reflexive: Themself (only obvious singular form)
I think it works well enough.
The only trouble is, it could be the top of a slippery slope. If we start allowing one relaxation of the rules, what is going to be next? I have a nightmare about "it's" becoming accepted as 3rd person singular neuter genitive, or "would of" replacing "would have".