People probably don't use it because at every English school in non English-speaking countries, they teach you that "they" is for plural only. I had personally never heard until recently that "they" could be singular too. It also really sounds weird to a French like me.
When were you taught to use "he" as a gender neutral pronoun on English? I've never heard anyone in an educational position advocate that.
I was taught to use the horribly awkward "he or she" (his or hers, etc) or alternatively to restructure to actually refer to plural. e.g. Instead of "A software engineer should test his code" use "Software engineers should test their code".
I grew up in the US, and I still think that singular "they" is weird. I remember thinking that it was weird in grade school, and wishing that there was a better gender-neutral pronoun.
Part of the problem with it being used in singular and plural context is that it's possible to use it in an ambiguous way where the ambiguity comes from the lack of enough context to differentiate between plural or singular. Separate words would remove this.
> It also really sounds weird to a French like me.
It sounds weird, and incorrect, to a native English-speaker like me, particularly when 'they' and 'their' are used more than once or maybe twice in a sentence.
If you don't know whether someone is male or female, use male pronouns until told otherwise. That's the rule in English, and it works.