The original question was whether other languages have "more elegant" solutions to the problem of binary gendered pronouns than we have in English. But the singular "they/them" only seems "clunky" because it's unfamiliar, and I maintain that this is something we'll all get over soon enough as we become more accustomed to it.
I'm not sure whether that's strictly correct. The point of different words, or rules about how words are used, is to allow nuances that are lost when those distinctions are surrendered. Often it doesn't matter, but sometimes it does.
So you can see how "Peter discussed with Mary whether they should take the new job" now leaves us uncertain not only which person, but even how many of them, is/are seeking employment. What's more, we're talking about third-person pronouns here. By definition, those are mostly used in the absence of the person involved because, if I'm present, I'm probably either "I/me" or "you". And discussions that happen when I'm absent are those most likely to involve people who don't know me and don't know my preferred pronouns, so have the worst chance of decoding that kind of sentence about me.
Whereas a new, non-gender-specific pronoun would at least have narrowed the field a bit.
Also, there's clumsiness from knock-on changes. For example, "If I wave to another cyclist,
are they likely to respond?" But there's only one cyclist and, even if we allow that the pronoun is they to be gender-neutral, why has the verb gone plural too? So we've got a complete change in the rules for conjugating verbs if, and only if, one particular pronoun is in use. That makes parsing sentences harder, surely?
What's worse, sometimes I think it's just clumsy because there's a more natural answer. If you're going to put the pronoun in the plural to avoid an unwarranted gender implication, why not just follow through and do the whole sentence? It doesn't always work, but often it does: "If I wave to other cyclists, are they likely to respond?"