'Single sex schools are not about creating a 'girl-culture' though, it is about education and growing up away from the male gaze.'
Single sex schools are about making 'girl culture' all about education. (Not sure about the 'male gaze' bit because that is there as soon as you walk out the school gates.) 'Girl culture' when it means a pov where girls are limited as to aspiration and participation is what girls' schools were set up to counteract back in the Victorian era. 'Girl culture' does not necessarily mean second class citizen culture. The assumption that girl culture is automatically going to mean something fluffy and insubstantial is sad to see.
I went to an all girls primary and mixed secondary myself. My oldest DCs went to a large American high school and DD1 will graduate from a large American university next year and go to work in a male dominated and maths oriented field. The culture of the wider society and of the family itself has an impact far greater than the colour of bike you ride when it comes to forming your opinion of yourself and setting your course through life imo. If you have chosen the bike yourself, paid for part of the cost from your own earnings, and used the bike to get to school at 5.30 am for swimming practice before school all through three winters then you are probably going to do well no matter what colour the bike is.
I don't know if I would go so far as to claim my DDs thought their bikes were better than the boys' bikes because it seemed to me during their childhoods that they existed in a world where competition was either amongst boys or amongst girls. They certainly thought their bikes were really nice, and expressed that in terms of being nicer than or as nice as the bikes of other girls. DS liked his first bike very much as it was like the bikes of the boys he was friends with (as opposed to the bike exH wanted to get for him. DS wanted a bmx and exH was inclined to get him a racer style). He hated the second bike because the handlebars were wonky and loved the third one, which he bought himself.
It was my observation of my DCs that they didn't compare themselves with children of the other sex. The DDs compared themselves to girls and DS to boys. Looking back, this was how I saw things in my own childhood too. It would have seemed like comparing apples and oranges to compare myself or my stuff to a boy or his.
Raven, men have a bathroom symbol too, a figure who is either stark naked or wearing pants, however you wish to interpret it. Not all the men of the world go around in trousers. Not all the men and women of the world could have the words for Men's or Women's Bathroom printed on the door in their own language. Bathroom door symbols are a convenient shorthand for a situation where a convenient shorthand is necessary. You could have a red door and a green door but then there would be the colour blind to consider. And there would be people saying that the use of one colour to mark out one sex made that sex 'other' and inferior, which seems to be what is happening here.