I was kind of surprised how many people can't settle on a specific gender
You could cut the gordian knot by borrowing Randall Munroe and Relsqui's solution for the xkcd color survey, which was to ask about chromosomal sex:
Do you have a Y chromosome?
[Don't Know] [Yes] [No]
If unsure, select "Yes" if you are physically male and "No" if you are physically female. If you have had SRS, please respond for your sex at birth. This question is relevant to the genetics of colorblindness.
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I'm not sure how any of these wordings of questions handle people with ambiguous genitalia.
Fair point. I'm not sure either; I think I'm relying on a given individual who is e.g. intersex either a) knowing that, and being able to make a better-educated guess about their chromosomes than any heuristic I offer, or b) not knowing that, which I'm willing to assume correlates well to having genitals that either do look like a penis or don't.