No, ladies and men's signs on public loos are not gender stereotyping, unless they are respectively pink and blue. No-one has ever suggested that describing a woman as a woman or a girl as a girl is gender stereotyping. HTH.
They don't describe men and women they quite often uses symbolism. For example a man is a stick figure and a woman is a stick figure wearing a dress.
That's awkward because many girls don't usually wear dresses.
You can protect your children from gender stereotyping without imposing it on others. Simply adopt it as part of your life instead of coming on these threads and preaching. What's does it matter to you what someone else believes? It's like the new secular evangelists
Dd at 5yo described a situation recently where she reached the story's climax and said, "then she screamed like a boy...!" and collapsed into fits of laughter. There was no irony.
That little girl has been raised so far to know that the only thing a boy can do that she can't is pee standing up if he wants and the only thing she can do that a boy can't is one day give birth to a baby if she wants.
To me that casual gender flip showed me that I was doing my job right and I'm meeting my responsibility to my child. We are all chilled, and it's okay.
No one seems to be happy that things are moving forward. It's like they're addicted to being angry and regardless of how quickly progress is made, it will never be quick enough for some. Wanting things to move too fast, or be coming pedantic, can be detrimental to a cause.