Singular "they" is gender-neutral and grammatically correct, I don't know why more people don't use it.
I remember getting points off of papers for misusing "they" in a singular context.
So it sounds weird to me too. We were always taught that "he" is a gender-neutral singular as needed.
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'm glad the "neutral they" is catching on.
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.
Though admittedly it's a corporate idiom that has passed its best-before date. Kind of like "folks".
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.