Transsexualism is a problem only because sexism is a problem. (Transgenderism is no problem at all: females have been wearing pants and so on for ages, and there’s nothing preventing males from wearing dresses and so on.)
If being female didn’t put people at greater risk of sexual assault from males, it wouldn’t matter whether male-bodied people were placed in women’s shelters and women’s prisons or allowed in women’s washrooms and change rooms. In fact, there wouldn’t need to be women’s shelters and separate prisons, washrooms, and changerooms. (Although there could be, for reasons other than fear of violence: there could be male, female, and mixed sex facilities across the board.) There may not even be a need for sex-segregated services; we would need only sex-tailored services.
If being female-bodied didn’t disadvantage athletes for most sports (as we have come to know them), it wouldn’t matter whether they had to compete against male-bodied people. In fact, there wouldn’t need to be sex-segregated sports.
If being female didn’t mean unjustified subordinate treatment, there would be no need for compensatory programs or data collection to monitor such treatment. And so it wouldn’t matter if male-bodied people skewed or eliminated such data collection (by making it illegal to record sex) or diminished the funding for such programs (should that be a consequence for refusing to serve male-bodied people).
In short, if there were no sexism, it wouldn’t matter whether males said they were females and females said they were males. Just as it doesn’t matter whether brown-haired people said they were red-haired (except maybe to a psychologist interested in delusion). (It would be a problem, however, if white-skinned people claimed to be black-skinned, because racism is a problem.)
Further, to the extent that transsexualism involves transgenderism, it depends on sexism. If not for sexism, there would be no need to change sex in order to change gender. If not for sexism, there would be no gender: the various attributes that are grouped together and then aligned with one sex or the other would be just individual attributes, as likely to be present in, or desired by, any given male as any given female.
 But there is sexism, so it does matter. By identifying themselves as female, and demanding access to women-only services and activities, ‘transwomen’ are oblivious not only to biological reality, but also to sexism. Or perhaps they are simply insensitive to women’s fears (which in itself suggests that they are not women, but are, in fact, still men). Because how can they not understand that someone with male levels of testosterone and male muscle mass is unwelcome in places where women would be vulnerable to their propensity to violence? Especially since there is much evidence showing that males prone to violence against women see nothing wrong with using deceit to gain access to women, and no evidence that males who are in various degrees of transformation are any less violent. (Of course ‘transwomen’ are also at risk of men’s violence, just as effeminate men have always been, but that’s a problem that men, not women, need to solve.)
So … until sexism has been eradicated from our society, ‘transwomen’ will just have to abstain from sex-segregated sports and wax their own balls (or, here’s an idea, go to waxing clinic that has personnel with experience waxing testicles—that is, a men’s waxing clinic). Is that too much to ask?
As for public restrooms and change rooms, if ‘transwomen’ are afraid to continue using the men’s rooms, they should lobby for trans’ rooms, not the right to use women’s rooms. As for prisons, I suppose ‘transwomen’ could lobby for separate trans facilities within men’s prisons.