Isn't stratified power along gender lines exactly what sexism gives sexism its power?
Not at all.
How can society be sexist without power stratified across gender lines?
Because we judge people based on their gender and prescribe behavior based on gender.
Race x Economic Class = :(
I'd certainly agree that race is a significant factor and contemporarily you can't extricate it from economic class. Just as you say, the effects of sexism are statistically enhanced when you apply them to women of color. They're also enhanced, though, when you apply them to men of color. Black men are, like men, overrepresented in prison, in the military, and on the streets, while being underrepresented in college.
'Three plus one' and 'two plus two' (sorry, Hubski formatting) in this equation are, as you might say, "two sides of the same coin". This is true. Does that mean, though, that only 'two plus two' is legitimate? That only 'one plus three' is legitimate? Regardless of how we frame it, how we break the numbers down, the point is that 4=4.
1 (plus) 3 = 2 (plus) 2
If that's a little less than clear, I'll be more explicit. We've got this dichotomy of agency/vulnerability, yeah? You can describe it either way. You can say that men are considered to have more agency, more culpability, and less vulnerability, or you can say that women are considered to have less agency, less culpability, and more vulnerability. These two statements mean the same thing, the only thing that's different is who we're focusing on as our object.
The only difference I see in what we're saying, as far as our description of sexism, is that an explanation supporting patriarchy theory insists on emphasizing one side of the coin over the other. So we'd say men may die in droves in the military and at work while women don't, but that's because we don't take women seriously. That doesn't do anything to deny the legitimacy of the statement, but it attempts to invert it to dismiss male vulnerability as objects of industry and war in favor of a specific narrative in which power is stratified by gender.
It's tricky but it doesn't really mean anything. Does that make sense?
We basically need a transitive property of sexism to clarify. The interconnected set of biases and double standards that make up the sexism faced by both genders (and thus, lest we forget, all human beings) is universally needlessly limiting.
Again, nowhere else in intersectionality do we see a supposed privileged class that faces any form of discrimination or increased statistics for poverty, imprisonment, mortality, or at-risk behavior. Nowhere. How are we to believe that this is an anomaly when it comes to men when we've got documentation of institutional and systemic sexism against men?