All languages have the resources to distinguish between the male and the female among animal species. It has, after all, always been fundamental to reproduction and until comparatively recently, to social organisation. So it is not surprising that languages have a male and a female gender. Some, like Greek and Latin, had a separate 'non-gender' (in both cases, the word for 'neither', 'oudeteros' <ουδετερος> in Greek and 'neuter' in Latin). AS has already been said, matters got confused as genders were used for the inanimate as well as for the animate. English is one of the languages that, though it does not have gender as a general feature of nouns and adjectives, unlike the European romance languages, and yet, unlike those and other languages, does have a neuter form of its 'personal pronoun'.
I am not aware of any language that has the resources to cater for multiple or binary gender identity. Biologically, the difference between make and female is clear, with comparatively rare exceptions. So the problem is how a language can adapt to the still controversial idea that there are some people who do not wish to be identified by one of the two 'standard' genders. The answer, I'm afraid, is that it cannot. First of all, the idea of non-binary gender is far from widely enough accepted to be integrated into everyday conversation or even formal writing. Second, there is no resource in the language for 'both'. You could use the term 'ambo' to designate the identity itself, but that does not supply a recognisable personal pronoun. There is none available.
So should we resort to the standard gender-avoidance strategy of the genderless plural 'They/them/their'? Well, we could. The trouble is that in that case we should have to become accustomed to the use of this generally plural pronoun as the immediate subject of a singular verb. But how can this happen by any natural evolution, when the English speaking world is far from accord on the distinctions that lie behind it? The alternative, the use of the neuter pronoun, which you mention, poses much the same problem.
If there is a solution, it is probably that the binary person's name will have to be used when their grammatical role is as subject of a verb, and 'them' and 'their' elsewhere. I am not proposing this: only predicting that that this is more likely to have linguistic legs than any alternative.