(Grew up with family that was highly conservative, and held such value systems.)
The issue, is that men need to appear "Strong" in the face of adversity. This means being the backbone that stands up to the soul-crushing realities of life, and forces that reality to realign in a fashion that benefits the family. There is no time to invest in exploring one's feelings or emotions-- The only thing that is of value is the decisive action; The commanding authority to make something happen, where otherwise it would not.
The flipside, is that the female role in the paradigm is to see to the emotional and mental health needs of all others in the group. (while being selfless about that herself, to parallel the man's emotional sacrifices for the family unit.) She is the emotional glue that holds everything together, while dad is the iron will that parts the sea of adversity.
When seen from this angle, the narrative about "Not crying in front of the children" makes more sense--- Dad is supposed to be unflappable-- Always able to take action, reliable, consistent, strong, and fierce. Crying on mom's shoulder in public view is the exact opposite of that. (It is heavily implied that Dad cries on mom's shoulder in private, away from everyone else-- At least in my flavor of this toxic regimen, it takes strong coloration from biblical prescriptions about what the ideal partnership is, and his is one of the things that is meant when the old testament describes the woman's role as the man's helpmate, and what is meant about the man being the head of the household; It undermines the projected strength of conviction and will, for the woman to publicly assert dissenting views. If desired, I can produce suitable direct bible references for this behavioral paradigm.)
You are correct that BOTH genders actively police and enforce the paradigm. Women (who are indoctrinated under it) are disgusted by "ineffective" or "weak" men-- Men (who are indoctrinated under it) are disgusted by "willful" and "prideful" women. Simultaneously, such indoctrinated women are aghast and shamed by being in the presence of willful women, and indoctrinated men likewise by being in the presence of emotionally supportive men, or men who show public signs of emotional vulnerability.
Rather than objectify it in an artificial crucible of "(dramatic music) Patriarchy!! (lightning crashes)", it is better to view it from the historical perspectives which gave rise to it, and the economic and societal conditions that fostered the creation of this tight feedback loop on mutual enforcement. Rather than focus simply on just one side of the suffering ("The suffering of WOMEN is unbearable!! Men have all the power!!", etc)-- it is important to understand that men suffer just as much under such a system, just in different ways. Sure, they have all the executive power in public, but they also have crippling emotional traumas they cannot express publicly, for fear of social ostracism or worse.
It has been getting more coverage in recent decades, but for the LOOOONGEST time, the notion that men suffered at all under patriarchal social conventions was met with spittle-faced hostility and incredulity by the feminist movement. (and is one of the reasons why I do NOT self-identify as a feminist, but instead as a gender egalitarian.) That it still does happen, is why I still do not approve of feminism's operative force. (the goal is laudable, but the methods matter.) Just bring up "Men's rights" some time, and see what happens. (Count to 20, and see how long it takes for the term 'misogynist' to come up, for instance. Nevermind the reality that just as it is possible to have a discussion about women's rights without misandry, it's converse must also be true. I view the assertion that it is not so, to be a demand to have the cake and eat it too, from the female perspective of the toxic paradigm-- Men are not allowed to have feelings, needs, or wants (in continuance of the toxic paradigm) but women should have executive power equal to them.) I find that to be wrong in every capacity, which is why I self-identify as gender egalitarian. All people are fallible, which is why all people need to take comfort in each other, and listen to the ideas of all of their peers openly.