Rather than going as linguistically far afield as Armenian, I could note that third-person possessive pronouns in most of the Romance languages encode the gender/number of the sentence's object rather than the sentence's subject.

This sometimes causes confusion among, say, French speakers learning English or English speakers learning Spanish – but French and Spanish speakers suffer no lack of clarity or precision when speaking their own languages and using those effectively gender-neutral possessive pronouns. (Neutral from the point of view of an English speaker who expects them to do different work than they actually do.)

Indeed, French, which I speak well, has developed all sorts of structures to make gendered communication precise within the pronoun system their language prefers.

And that raises another interesting point. I'm all for decency and respect, and I'm happy to use they/them as gender-neutral pronouns for non-binary people and people whose gender I don't know. I recognize, however, that if this becomes a natural part of our language, we will adapt naturally (unless some vast sociological shift in gender views also happens) by shaping the language to specify gender in other ways.

Languages are wonderfully adaptive. There's really nothing to fear.