The first instinct, for me at least, when considering that a simple binary model of gender is insufficient, was to just say "Hey, we'll just make a gender continuum!" The thought here is that you would have two polar opposite points on a continuum--archetypical man and archetypical woman--and a person's gender can be located anywhere along that scale in between those extremes.
This might be helpful for those "hard cases": someone born with ambiguous genitalia might be right about in the middle of the continuum, effeminate men would be (as the adjective implies) closer to the middle but still on the "man" side, and other cases might be resolved similarly. Case closed!
Not so fast, though. Who decides what the extremes are? As my picture shows, is manly man required to be super-buff and a protector while a woman has to cook and care for dozens of kids?Are those really what men and women fundamentally are? This is just the same problem as the basic binary model: we still can't define what must be there for a woman to be a woman or for a man to be a man. A single woman who has no children cannot in any rational way be called less of a woman than the mother of ten.
Even if we could define the prototypical man and woman, a continuum is still too limiting because it assumes that you can rank every person as either more manly or more womanly than anyone else, and that is just ridiculous--if two men are identical except one loves football and has a high voice while the other loves opera and has a low voice, how do you rank them based on their differing attributes? They send such mixed messages! It seems pretty clear to me that a continuum model for gender is only slightly better than a simple binary model. So if that doesn't work, what will? I have a possible idea, but it will have to wait for Part IV.