No, it’s not. I’ll continue my explanation by responding to the rest of your post.
Here is where you make your mistake. You assume that the alternative to “women” is “men who think they are women” and that’s not true, even though a few of your opponents have used that phrase. Leaving aside the word woman, trans people are not claiming to be biologically female. They are claiming to share the social identity commonly associated with females.
What is important is not the word, but the meaning. So instead of the word you define as the typically female social identity (women), we can use another term, say “group A” instead. “Trans women are group A” now means the same thing as your concept of @trans women are women” with your definition of woman.
However, historically women has not meant “the social identity commonly associated with women.” It has been a synonym of female, or as some have posited “adult human female”. In order for “ trans women are women” to be true, the term woman has to be redefined. But the concept can be true without redefining woman. If we don’t redefine woman, “Tran women are group A” can still be true.
Group A would be made up of trans women and cud women just as you would say women is made up of trans women and cis women. You will note I used the term cis women here. I did so for a different reason than usual. If we don’t redefine woman, then trans men are still women but they are women who share the identity group B with men.
The redefinition of the term woman is an area of dispute. But the concept represented is less controversial. It is possible to support the concept but not the terminology.
Now obviously, we aren’t going to say group A or group B. I’ve used the term “people who identify as women” which isn’t really accurate either. Maybe a more accurate phrase without redefinition would be trans woman and women share a female identity. Or something. It’s not as cool a phrase, but it doesn’t change the meaning of things like “Women’s health Center” either.
As I illustrated above, it’s possible to separate gender and sex without separating women from female. In order separate woman from female, though, you have to redefine woman in terms of gender instead of sex.
There is, obviously resistance to this. You may interpret this resistance as anti-trans. And in some cases you would be correct. But not necessarily. It is also possible to support the concept of a shared gender identity without offering up “woman” as the name for that gender identity.
As I’ve said “woman” is a word that holds a lot of meaning for people on both sides. The power human adult females invest in the word has nothing to do with how they feel about trans women.
The term bigot is thrown around a lot. And yes a bigot might suggest that homosexuality is the result of a psychiatric or genetic disorder. But a scientist investigating the cause of homosexuality might also put forward those hypotheses without being a bigot.
Also, terminology for medical and psychological issues changes. Sometimes not for technical reasons. Terms that are descriptive enter the common lexicon and become insults. Retarded is a case in point. Disorder is not a bad word.