But they took a different route and actually invented a new word for the non-identified gender of person. So instead of hon (she) or han (he), hen is used (note that it not "neutral" - that is det (it)). It has been taken up in the media and people use it with no fuss (very Swedish - just get on with things!).
They had it easy though - an ungendered language, and they could follow the lead of their superior Finnish neighbours.
But pity the poor Finnish gender-specials. Finnish only has the neutral "hän" (he/she), or "se" (it). ("Se" is actually used more for people than "hän" in common speech in most dialects.)
How is a non-binary supposed to signal their non-binariness when everyone is using gender-neutral pronouns?
Or, indeed, how is a trans person supposed to indicate their opposite-genderedness? Complaints of "misgendering" due to lack of gendered pronouns occur. English speakers can pick up that someone is supposed to be a "transwoman" by people using "she" about them, but I can imagine there's more scope for transness going unnoticed in the absence of people dropping pronoun hints.
Apparently it's not uncommon for gender-special Finns to list their English pronouns. LOL. It's the only way they can try to make pronouns a thing.
And they have to try to make it a thing because it's one of the rituals of the group belief system, even in a context where it makes no sense whatsoever.