From your article:
By the middle of the century, scientists set their sights on reaffirming the ideology by undertaking research to prove that the female body was inherently weaker than the male body.
Armed with such “scientific” facts (now understood as merely bolstering political views against the emergent women’s rights movement), legislators and other policymakers began enacting laws aimed at protecting “weaker” women in the workplace.
I assume this is the crux of your argument about why separate spheres are bad? The fact that old-fashioned scientific thinking was used against women in the past does not negate the fact that humans are a sexually dimorphic species. Women are not dumber or less evolved than men, as previously believed, but we ARE physically weaker and smaller. That is ABSOLUTELY scientific fact and the same is true of many other large mammals. In fact, the only reason scientific thinking of the time had a basis for sexist theories was because of observable physical differences between the sexes. If women and men were the same height and had similar skull sizes, they couldn't have made that argument. Men are also more likely to commit sexual offences than women and the vast majority of men are attracted to women.
Pitting sexual assaults in mixed sex workplaces against sexual assaults in public restrooms is disingenuous because most public restrooms are single sex or a single stall. If women experience fewer sexual assaults in public restrooms, it will be BECAUSE THEY ARE SINGLE SEX. I fully expect men and boys are far more likely to be sexually assaulted in men's restrooms and changing rooms than they are at work, hence why women parenting alone in public often bring their young sons with them to the toilets or wait outside the men's for them but within earshot. It's an enclosed space where men partially undress, that may be quiet or deserted for fair stretches of time. Ten minutes is more than long enough to commit a sex attack.
Regarding the trans issue, every trans woman could be an angel sent from heaven, made from fairy dust and the loudest and most majestic of unicorn farts. Every one of them could be a good, law-abiding citizen with no malintent towards women at all. The fact remains that if you tell the general public that anyone can get into an enclosed women's only space just by wearing women's clothes and claiming they are a woman, then you will get male sex pests and predators dressing up as women and declaring that they are trans when they aren't. Good men don't want to get in, they have no motivation to and are happier to observe social norms and women's comfort. If I announced in a national newspaper that anyone who dressed like my partner and claimed they were my partner could have access to our joint savings account, what sort of person do you think is going to take me up on that?