IIRC, “man” in the sense of “human/person”, from which comes “mankind”, is actually older than “man” in the sense of “adult male human”.

> Why not "people", "humankind", etc?

That's what “mankind” has always meant.

You can always use a different word if you prefer to, but please don't claim that e.g. Neil Armstrong did or wanted to exclude women or children when he said his walk on the moon was a giant leap for mankind.