That’s not “the arguments falling apart” that’s you choosing to cling to an ideology that is insane to any thinking person. Presumably because you think lying in this way makes you a good person. Or maybe because actually solving problems takes more thought than you’re willing to put in.
Transgender people are not gender ideology. A criticism of the latter is not an attack on the former. Gender ideology is an embarrassingly disingenuous attempt to deny reality by twisting the meaning of words. Transgender people are human beings with gender dysphoria who, just like everybody else, deserve to live their lives in safety and dignity.
Unfortunately, just like everybody else, that safety and dignity has to be balanced with the rest of society's. And the problem with you ideologues is that you reject any attempts to find this balance. The only acceptable solution seems to be that the entire and ever-expanding "trans umbrella" get all of their demands met, no matter how unreasonable, and women should just shut up and accept it. Even if failing to differentiate exposes women to danger and discomfort in the process. After all, “nuance is an excuse for inaction.” Right?
So while I can happily offer solutions to the problems I've pointed out (I've written a few times about potential solutions to the problems I've raised here), those solutions mean considering the feelings of women, something you seem utterly unwilling to do. And it means recognising that sex is more meaningful than gender identity in most cases of sex segregation, something that should be obvious from the "sex" part of "sex-segregation."
But when you ignore women’s right to safety and privacy, when you demonstrate that you don’t have even a quarter the compassion for women that you have for trans women, you make life harder for women AND trans people. Because it’s clear to anybody who’s paying attention that more and more of the public is conflating ideologues like you with ordinary trans people just trying to live their lives. And when they hear what you say, they stop trying to empathise and simply decide that you’re all cultists who need to be stopped. Maybe that’s not your intention, but I want you to be aware of the result of nonsense like this.