I think a lot of the problem arises because we confuse 'sex' with 'gender'. While gender is now accepted as a societal construct, it is originally derived from one's sex.
So sex = female and male (fixed binary biology)
gender = woman and man (fluid and varied)
Up until recently, female = woman and male = man. For the vast majority of us, this is still the case.
Transgenderism argues that you can now choose your gender according to how you 'feel' inside. Which is fine, if rather confusing for most of us. The problem arises because males who wish to be known as women, have neither the same 'sex' experiences as females e.g. female puberty, periods, childbirth, lower athletic abilities, poorer musculature, shorter stature, less aggression etc. nor the same gender experiences e.g. being judged by appearance, greater clothing choice, higher caring expectations, lower pay, etc.
This is a gross generalisation, as there is a wide variety between both sex difference and gender. But the sex differences hasn't changed all that much. The problem arises because the role of women in society HAS changed and is highly varied across different cultures. For example, I don't know of any transwomen who would want the role of a women in a society where women are highly restricted in what they can do or look like.
The truth is that, at least currently, you cannot change your sex. You can ask to be treated as the opposite gender, but you cannot become the other sex. That would be like asking to become an Elephant or a Penguin. You could look like one, at least in part, but you cannot become one.