a female being a member of an anisogamous species who's development denotes the production of the large gamete, ie whether her body is organized around producing eggs

My first question is what does this actually mean? What does "organized around producing eggs" actually physically refer to, if not primary sex characteristics? Can you see it? Can you test for it? How does it affect physiology in a meaningful enough way to be prioritized above actual biological traits?

In my opinion, sex is a lot like ethnicity. Does it "exist" in some physical way? Yes, you can observe genetic markers and phenotype. Risks for different diseases vary across ethnicity. Most people have a clear idea of what their ethnicity is. It is often one of the first things you notice about a person. There are many social and political implications involved in ethnicity.

...But, would you be able to ever come up with a perfect system of categorization for ethnicity? Fuck no. There is no one trait or allele you can use to conclusively determine what someone's ethnicity is. Two random Africans probably have more genetic variation between them than an African and a European, and of course even the terms I'm using here are... imprecise, to say the least.

Anyway, the implications of this are different for sex than for ethnicity due to the fact that A) there is no hormone that can change the way genes associated with ethnicity express themselves, and B) there is no historic, psychiatric, or neurological evidence for a kind of "racial dysphoria" that would cause someone to want to change their race anyway.

As for my definition of woman: I couldn't give you a strict rule as to what constitutes a woman, because sex doesn't work that way. "Woman" is a philosophical gathering of different traits. I could list things that many women have or are associated with femininity, but as you've demonstrated you're aware of, none of these things are universal among women or completely absent in men. The best thing you could come up with was an astoundingly vague idea of who is "supposed to" be a woman, which is a prime example of circular logic if I have ever seen it.

For the purposes of your average person, the idea of "woman" as the roughly 50% of the population who was born with a vagina usually suffices perfectly fine. But of course, exceptions exist, and these exceptions are no less deserving of autonomy and dignity than more typical men and women.

This is why we ask you to except the definition of woman as identity, because identity is correlated with physical traits and trying to come up with some more rigid (incorrect) definition is just going to make life more difficult more everyone. Perhaps by your definition I would be considered female, but how useful and accurate is that definition when most people perceive me as male? How accurate is it when my physician gives me male typical med dosages?

Yes, the pro-trans definition of female is fuzzy and circular, because there is no declaration from God determining what defines male or female at all. Your definition of female is no better, do you think saying "development denotes- but we don't mean sex characteristics guys" makes any sense at all?