This was such a great read. I am not a linguist, let alone a philologist, but I had this thought.
The modern use of "gender" to mean social aspects of sex (or sex) and the historical use of "gender" to mean sex are still actively used. In a very real sense, these are exclusive and oppositional categories, so any ambiguous use of "gender" in contemporary English should be interpreted as auto-antonymal use. That is, depending on what the speaker means, their meaning could be interpreted in exactly the opposite way.
Auto-antonyms are rare in English, but the obvious ones (https://en.wikipedia.org/wiki/Auto-antonym#Examples) do not cause confusion. I would suggest that outside of specific professional settings—philosophy, biology, medicine, linguistics—"gender" is almost exclusively an auto-antonym. Perhaps this contributes to the nearly impossible discourse on the subject.