The toilet question is quite the red herring.
Toilets have been separated into male and female facilities for decades for many reasons - some for safety, some for convenience. For example, it is much more efficient to have a row of urinals for men to use, but it does mean exposing their genitals so for their dignity they are given a space understood to be single sex for them to do so, and cubicles for anyone wanting further privacy or - ahem - anything else other than a wee.
Were anyone - male of female - standing opposite the row of urinals ogling penises in a predatory way, they would be challenged, and hastily ejected. Can you see how much easier it is for men to police that?
Can you also see how if a predatory woman positioned herself at a urinal, maybe utilising a she wee or simil3, under this current climate it would be harder to challenge her and have her appropriately sanctioned? She could, after all, just claim to be a trans man - which requires nothing other than to say 'I am a man' even if she is wearing a full face of make up and knee high boots and doesn't actually have a penis - and it would be her accusers facing the full wrath of the law for being transphobic.
Now think of a ladies loos. Think of the extra things ladies need loos for, and how most of that requires pretty much stripping from the waist down. And how much of a difference it makes not being able to challenge the guy in the cubicle at the end, and get nearby security to escort him safely away......