I am a trans woman, I see myself as a woman, yet I find no fault in this assessment of the current situation, except in analyzing it in terms of Progressivism. I think that's misleading.
There is a resolution to the tension that the writer discusses; but it won't be popular, which is to say that it will not be "Progressive." It will be conservative, in the most basic sense of the term.
Sex is a fact, while gender is a belief. It cannot be proven that gender, as that term is used in (though not defined by) the trans community, even exists. Maybe it does, maybe it doesn't. And because gender is a belief, being transgender is likewise a belief. Maybe trans women are women, maybe they're not.
In a free society, we do not base law, public policy or civil rights on mere beliefs. To do so would be like enshrining into law a tenet of Islam or Roman Catholicism.
Who is a woman? No one would argue that Kris Jenner isn't a woman; but whether Caitlyn Jenner is a woman is a question upon which people may reasonably differ. Thus, Kris Jenner should reasonably have access to women's safe spaces and women's sports, and, if convicted of a crime, be placed in a woman's prison; however, there is no proof that Caitlyn Jenner is similarly entitled. When the Progressives come claiming that Caitlyn has a right to these things because she's a woman, the response is, "Prove it." The screaming won't end there, but the argument does.
Of course, none of this means that trans women don't need a place to use the bathroom safely (we can't use the men's), or aren't at high risk in men's prisons. We do, and we are. I cannot travel safely anymore, and when I leave my home I have to know I will have access to a single-user restroom somewhere or be able to return home if I need to potty -- something cis women haven't had to worry about for over a century.
But that doesn't give me an enforceable right to use women's spaces. I am, of course, always touched and happy when a cis woman invites me to do so; but I don't always accept. I know that not all cis women will welcome my presence, and the last thing I want to do is make someone afraid.
Fights that gay women fought in the days of Beebo Brinker are being fought by trans women today. But there are not so many of us.