Either way, Tolkien's pun only works if the reader was similarly misled by the same prophecy; that is, even if the reader is aware that Eowyn is a woman, they may be unaware up until this point that the prophecy about men does not apply to women (whether because they assumed a gender-neutral usage of "men", or because the possibility did not occur to them). This problematizes the interpretation of Tolkien's usage of "men" as gender-neutral, since Tolkien almost satirizes this linguistic ambiguity.

There is another reading of this passage which says more about lingual ambiguity leading to a prophecy being both true and false at the same time.

"The Witch-king's boast that no man can kill him comes from the elf Glorfindel, who foretold "not by the hand of man shall he fall." However, this line was always more of a symbolic prophecy than literal confirmation that men physically couldn't slay the Witch-king. Éowyn and Merry's victory fulfilled Glorfindel's prediction, but their gender and species weren't the deciding factors in bringing down the Ringwraith, it was a unique alignment of fated circumstances, which as Glorfindel predicted, didn't involve a man."

I don't know the circumstances around this prophecy, but it's entirely possibly Glorfindel was in fact relying on the ambiguity to deliberately shroud his vision, knowing his words would be interpreted one way while still being technically true from another perspective. What I'm saying is this wasn't some magical law writ large that would've been understood without a dictionary.

On the other hand, I might assume this prediction was originally made in Elvish and it seems unlikely that both languages would share this particular quirk. Indeed, Tolkien being a linguist makes it even more unlikely.

This is all just fodder for discussion, though. I admire your effort and explanation. Tabletop gaming has gotten considerably more inclusive over the years thanks to efforts such as yours.