What you're talking about is the feminist theory of "Patriarchy hurts men too". It's not a new concept to contemporary waves of feminism, it's been around for decades, and it is crucial to these waves.
People who don't subscribe to this theory are bad at feminism. Just like they're bad at feminism if they disregard the plight of trans folk, or other Gender and Sexual Minorities as irrelevant. Just like they're bad at feminism if they disregard how race amplifies non-racial inequalities. These are all things that appeared in Second Wave feminism. That doesn't make earlier feminism irrelevant, it just made it less effective. It maintained various status quos in the facets of power that exist in society. arguewithatree points out that this has a name; it's called Kyriarchy. Patriarchy is a benefactor of Kyriarchy.
Sexism goes both ways indeed, and it won't be gone until the patriarchal imbalance is, with its benevolent sexism encouraging the demure domesticated woman and the strong industrial/warrior man. Tradition they call it. It is rooted in biblical text and everything. It sprouted a culture where men protect the virginity, faithfulness and safety of their "possessions" from other men. I can assure you the people who formulated this society based on biblical values weren't women. They weren't allowed.
These people in power, they sure as hell aren't going to listen to the women whose agency they've stunted, lest they lose even more control. And they're sure as hell not going to give vulnerability to men, because they don't want to be vulnerable themselves. They police masculinity just as much as they police femininity. They discourage men from pursuing nurturing careers, and outright banned women from combat roles in the military. They privilege the successful masculine man, and punish as "bitch" the ambitious entrepreneurial woman. They privilege the submissive feminine woman, but scold as "girly " the man who rejects the traditional masculine framework. They teach men to fear being labelled, and women to fear men.
It is people in power who are denying men's vulnerability and casting them aside if they are vulnerable. It is people in power in power who are keeping agency from women. It is people in power who are setting the standards for society. It's not sexism to point out that it is overwhelmingly or ultimately men who are in power, or that because of this they need to lead change or share the power with those who will.
In an amazing coincidence, as patriarchal dominance is slowly eroded, there are shifts in perception, barriers are being broken down, and things are changing.
"Patriarchy theory" isn't a feminist theory. It exists almost exclusively as a concept in anti-feminist arguments. It is designed to discredit feminism by twisting logic and painting a straw argument, a "straw feminist" so to speak. It is used to imply that feminism is about are running around screaming "sexism" and "privilege" every time they don't get their way while ignoring that men sometimes have bad luck too. Are there people who identify as feminists who do this? Sure. Are they the majority and is it the basis of feminism? Hells to the no.
Feminists don't expect men in homeless shelters to overturn sexism. They expect the privileged men and women in politics and the courthouse to.