In short, the enemy is the patriarchy, and if you think for a minute that that only negatively affects girls and women, well, then you've never been a shy nerd asking out that cute girl from your math class who's also a cheerleader. Spoiler alert, P(that goes well) < 0.01.
Certainly those ideals are rooted in sexism and are in many ways sexist, but the sexism isn't what caused the problem in my example. The gender norms by themselves (ie doesn't matter whether they're the way they are due to sexism) caused the problem, because they basically say that a guy is worthless if he isn't a nearly perfect example of the masculine gender.
She's rejecting him because he doesn't have enough of the traits that society says a man should have (and maybe has too many of the traits that society says women should have): He's not strong, he's not assertive, he's not domineering, he's not good at arbitrary athletic competitions, etc. etc. etc.
That's sexism.
"The first important thing to understand about sexism is that it is systemic. Gender-based prejudice, on the other hand, is individual, although it exists in the context of a sexist system."
"But men who experience oppression based on the way they express their gender are still not oppressed on the basis of their gender itself. That is to say, they’re not experiencing that oppression because they’re men. Non-men, on the other hand, experience gender-based oppression because they aren’t men."
[1]: http://everydayfeminism.com/2015/01/sexism-vs-prejudice/
This has everything to do with his sex. He is being rejected (hypothetically - within our thought experiment here) because he is considered to be an insufficient exemplar of the male sex. To phrase it a different way - he is insufficiently masculine for this cheerleader (i.e. a very high-status female) to have an interest in him.
You correctly point this out later on - She's rejecting him because he doesn't have enough of the traits that society says a man should have (and maybe has too many of the traits that society says women should have): He's not strong, he's not assertive, he's not domineering, he's not good at arbitrary athletic competitions, etc. etc. etc.
Sex is purely genitalia. The two extremes are male and female, but this is a spectrum and not a binary.
Gender is all the other shit that society piles on top of that. The two extremes in western culture are masculine and feminine, but this is a spectrum and not a binary. That this mostly lines up with the sexual spectrum is an arbitrary decision by society; society could define gender in such a way that it is orthogonal to sex. Examples of where gender doesn't line up with sex in western society include butch within the lgbtq community: http://en.wikipedia.org/wiki/Butch_and_femme. Other societies have in fact defined gender in a way that doesn't line up with sex, so this isn't just theoretical: http://en.wikipedia.org/wiki/Gender_identity#Non-Western_gen....
In the thought experiment he is not being rejected because of his sex. He is being rejected because of the gender he presents is not sufficiently masculine enough.
What does this is gender roles and their consequent beauty norms. People decide that women are best off looking fit and have big boobs and are outgoing and have a certain personality. These norms are communicated to them all throughout their childhood through media and peer pressure. In exactly the same way, ideals create the jock as well.
When you actually look at the people embodying these stereotypes, there's nothing all that much different about them, they're just people who've managed to adopt these norms. They might even be meaner than normal, enforcing these roles themselves.
Gender roles, beauty norms are part of the culture that feminists call patriarchy. Patriarchy is just sexism at the social / cultural scale. They wouldn't exist without these broader cultural dynamics, people would just be people and you could ask out whoever you want and she wouldn't shut you down because she has a place to maintain in the hierarchy, a hierarchy created and maintained by sexism.
Once you realize that women perpetuate patriarchy too and how, that's when you start to really understand just what it is.
If you need an analogy, think about how the general lack of respect for privacy at both the individual and the legislative levels negatively affects all of us, even those who 'have nothing to hide'.
I'm not really clear on how such divisions emerges from sexism. At its root, it seems to come from primate status competition, within separate groups that happen to be divided out by sex.
Among the boys, the jocks are those who prioritize sports and physical achievement, while the nerds are those who prioritize academics and mental achievement. This divide persists even in the absence of females - for example, all-boys schools still have aggressive athletes and quiet, bookish introverts, regardless of the presence or absence of the opposite sex.
If anything, the thing you're describing as patriarchy and/or sexism is the effect, rather than the cause.
you've never been a shy nerd asking out that cute girl from your math class who's also a cheerleader.
Feminism is designed to make sure it's illegal for the shy nerd to ask out the prom queen. Spoiler alert: feminists are not the friends you may think they are.
False accusations of any crime are horrible, but they are in no way common: The United States Justice Department agrees [with the rough percentage of false rape claims], saying false accusations "are estimated to occur at the low rate of two percent -- similar to the rate of false accusations for other violent crimes."
Coincidentally, 2% is the number of rape accusations that ever result in a conviction: https://rainn.org/get-information/statistics/reporting-rates (this sources back to the DoJ as well. RAINN just has a nifty infographic.