'On the contrary, there is no "Men's chess". There is chess, and there is women's chess.'
Wow, that's so depressing. 
This is the sort of thing that gets to me. I'm sure it's a great idea given that women have been pushed out, but it's also going to make women's progress slower. You will have a situation where little girls see chess as a 'boy's game'.
I remember noticing this. My dad taught me and my brother to play chess. For him, there were chess tournaments arranged through cubs, so it was very easy for him to play competitively at a friendly level. And he got quite good.
For me, sure, I could have played competitively, but it'd have required actively seeking out tournaments. Which would have been much more stressful and would have required more confidence.
I hope this has changed, but I suspect not enough.