A lot of this ideology is, I believe, based on the people pushing it being confused and misinformed and, due to their strong need for some sort of self or societal appreciation being unable or unwilling to discuss or consider their position.
Yes woman is socially constructed but really every concept is so that is really no reason to throw away all attempts to have a useful and largely shared definition. A concept without a definition is not a concept or a construct and in the case of 'woman' it is a very clear construct as it links 99.9% with one's chromosomes which are expressed through highly visible secondary sexual characteristics. Feminine is much more of a social construct with much less clear definition. It is therefore a much more contested idea and if transwomen, for example, were to say they identify as more feminine that their own personal internalised idea of masculine, I think they would find a great deal more support.
The social constructs we make have real-world effects. Women have been subjugated for generations based on our construction of woman. To deny the correlation with chromosomes does not stop the subjugation. It merely pushes it underground. Race is a much less clear construct as there is obviously no real clarity re: when someone is white or black (for example). E.g. Megan Markle considers herself black (understandably with a black mother). Her son might also do so. If he has a child with a white woman maybe his children will also. If those children have children with white women, would they also be black? How about if this continued for another 10 generations? Despite the inevitable wooliness around the edges, people do not want to do away with any shared definition of black because to do so would remove societies ability to recognise its prejudice and try and do something about that. As an example, the news is full of reports of how much more often black people are stopped and searched. If we do not have some shared definition of black (even if this can never e 100% clear) then we do not have any way of knowing this fact. And we should know this fact as without that knowledge we cannot own our societies institutionalised racism.
Saying something is a social construct is not a get out of jail free card to not attempt any meaning definition. Nor is it an excuse to hide societal prejudice.
It is shameful of any man - who, by definition. has grown up as the most valued sex and had experiences in line with that which shape them - to identify as a woman. If they want to identify as a transwoman, then fine by me and most of the rest of us. But identifying as a woman is a clear statement of lack of interest in women's oppression.
When we talk about other oppressed groups, we understand the need for the oppressed group to speak for themselves. This should also be the case for women. A transwoman's or a man's thoughts about how woman should be defined and how women's spaces should be protected is no more valid than my (as a white person) thoughts about what might be the best way to address institutional racism. I speak from a position of power and so my voice is rightly invalid. I'm not saying that all women do not believe that TWAW but I believe the vast majority do. If female TRA believe that they are actually in the majority, then given that they are trying to redefine the definition of woman (lets not pretend otherwise, that's clearly very silly) then it is
up to them to robustly prove that the majority of women support this change. Bullying, insulting and harassing women should not be their methodology of choice.