Just because a standard of treatment isn't codified in law doesn't mean it isn't structural. There are many sexist social practices that continue to be perpetuated and continue to hold women (and, by proxy, men) back. It does boil down to the same assumptions of agency and vulnerability but that doesn't make them any less real.
I'd certainly agree that there are many sexist social practices still in effect. What I take issue with is the presumption that women are the primary victims here or that when it comes to sexism men are only being held back, as you say, by proxy. We don't live in anything like a post-sexist society. Our society is still quite sexist, but it's not sexist in a way that stratifies power along gender lines.
This is where I take issue with the surrounding of patriarchy theory with intersectionality in order to, as I see it, more or less prop up this idea of cleanly stratified power dynamics when it comes to gender by comparing it to all these actual very clear stratifications of power.
For example, what I think is the most important intersectional axis (and unfortunately the one that seems to be most ignored), economic privilege, is incredibly clear cut. You don't gain anything special by not having money or lose anything by having money. Wealth is great for everybody and there's a clear disadvantage that comes with not having access to it.
Sexuality is another one that we can pretty easily see one group has a clear advantage over the other and there are no roads going in the opposite direction. You don't get a leg up on straight people in any way by being gay or bi. The same can be said of sexual identity. Trans people don't get a leg up on cis people. No ambiguity. Even straight people crashing gay bars are told to fuck off at worst. Maybe the supreme court decides down the road that you have to sell a cake. Nobody's beating the shit out of you or making up special words because they hate you. This is easy to see.
Race is a little less straightforward. Where do Asians fit in on our privilege scale? Are Jewish people white? What's the deal with Rachel Dolezal? I think if you have a remotely coherent picture of American demographics you'll realize that something's wrong with the way we treat black people, but that's about the only thing that's clear about race. Things are murky here, partly because a vague combination of skin color and nationality that has nothing to do with genetics is a weird way to classify people. At any rate, we're still in the territory of at least being able to identify a clearly disadvantaged demographic in the United States at the very least. Being white doesn't carry special disadvantages.
So far, I'm on board with intersectionality for the most part. I do think there are some people who focus excessively on race, but there are also people who focus excessively on identity politics related to sexuality, sexual identity, and even economic status. Somebody's got to keep the lights on at Tumblr. Anyway.
Gender is a whole different ballpark. Instead of a clear cut situation where one group is at a clear disadvantage to the other, you've got this whole interconnected series of double-standards. What's utterly unique here is that they go both ways. Nowhere else in intersectionality do we see this. Not a single instance anywhere. I mean, when it comes to race there might be instances when you're at a greater risk in some fairly narrow situation if you're white in the US, but nothing systemic, nothing institutional. That's not what we see with gender, though. When it comes to gender we have visible systemic and institutional sexism against men as well as women.
Intersectionality, as long as its used as a defense of patriarchy theory, will primarily be about propping up this, I'd say, rather flimsy narrative of stratified power between genders by associating it with legitimate instances of stratified power. That's where the whole idea falls down for me.
As far as getting anti-sexism to regular people, I'd say the best way to go about it is by being straightforward. We ought to treat everybody the same and that's all there is to it. We should focus on equality rather than the half-baked oversimplified versions of theory that you yourself suggest are likely to come across. I'd agree that the academic side isn't as important to individuals, but I don't think current antisexist campaigning reflects that. To me it reflects everything about patriarchy theory, and thus the continued underestimation of women and emphasis on men as stereotypical stoic protectors or aggressors.