Gender identity refers to how an individual experiences their "I". Nobody else but the individual him/ herself has access to a person's "I'. This "I" which we all have is the most fundamental experience of who "I" am. Absolutely nobody else can tell us how we should experience our sense of "I' because it is a subject and falls outside the domain of objective inquiry.
This right here is the root of the problem.
Transgenderism is acutely individualistic and does not take into account the fact that we live in a society.
"Women" is a word for a group of people. A group of people consisting of roughly half the global population. You are not a woman because you have decided you are one based on how you feel inside. You are a woman if you share the same characteristics that other members of the group share. The only characteristics that all women share are being adult, human and female. And female is NOT a "gender identity". Gender identity is made up nonsense that didn't exist until about five minutes ago, and most people do not have a gender identity in any meaningful sense. No, female means exactly the same thing in humans as it does in other animals. It means you are a member of the childbearing sex. You can still be female if your female reproductive system does not work properly and you cannot in fact bear children, or if you choose not to have them. But you cannot be female if you were born with a penis and are (or were potentially) capable of fathering children with your own sperm. Because that means you are male, which is the opposite of female.
Trans people's "lived experience" is completely and utterly irrelevant to the definition of "woman" or "man" because these terms are defined objectively, not subjectively.
I can listen to a biologically male trans person talking about their lived experience all day long but absolutely nothing they might say is going to change my understanding of what a woman is.
It is extraordinarily arrogant to redefine half the human population, who used to be defined as being members of the female sex, as people who have a particular gender identity that you believe you share.
Trans people are constantly saying that nobody gets to tell them how they identify.
Fine. Most people are not actually all that interested in how trans people identify, for the same reason that I am not all that interested in what my neighbour's favourite colour is or what takeaway my boss likes to get on a Friday night. Because it's completely irrelevant to our lives.
But by saying that a woman is a person with a female gender identity, you are actually telling half the world how we identify. You are telling us that a woman is something we choose to be, rather than something we were born at random. You are including us in your gender identity whether we agree or not. And then you are telling us that we cannot have any spaces or sporting categories which do not include you because you believe that you identify as one of us and we have no right to disagree.
Oh and by the way, sex cannot be changed. Having your penis amputated does not make you female. If an archaeologist in the year 3000 were to dig up the body of a post op trans woman, they could easily and correctly be able to identify it as the body of a male person.