People in general are hardwired to protect women, much in the same way people will protect children.
The grain of truth is that people in general tend to feel more empathy towards women than they do towards men, and our visual media reflects that. The faceless bad guys who get blown away in an action movie will typically be men, while horror movies that want you to feel something when a character dies will often use women in that role. That empathetic difference has nothing to do with some "hardwired" desire to protect women, it it has to do with the perceived relationship between gender and emotion in the society we live in.
The dominance of men over women is not rooted in the demonization of women as a class in the same way as other marginalized groups have been demonized. Women have, at various times, been viewed as more refined, more civilized, more beautiful, more lovable, more compassionate, more intuitive, more gentle and more loving than men. Even today, many "traditional" men will happily concede all these qualities.
In the pre-modern past, things were very different, women were viewed as inherently evil and morally weak and the idea of male beauty was more readily accepted, but in Europe at least that mostly disappeared sometime between the 18th and 19th centuries. The foundation of male dominance is not based on the complete negation of all positive qualities of womanhood, it is based on an idea of sex complementarity which ascribes to women superior virtues only in areas which do not actually grant them any possibility of agency.
It is normal in our society to love women, it has been for hundreds of years, it is considerably less normal to allow women any kind of control over their own lives.
The reason men do not elicit empathy is because the society we live in understands men as instrumental beings. The kinds of emotions that would elicit an empathetic response are far harder for a man to display appropriately without undermining his perceived instrumentality.
Are you saying the idea of women and children first off the Titanic was just rooted in patriarchal sexism?
Yes, it absolutely was. What else would it be rooted in?
This is a time when the prevailing scientific belief was that women were neurologically or psychologically incapable of reaching adulthood, and thus essentially were
children. Men were not expected to stay behind because of some intuitively understood biological imperative, but because the Victorian culture that came up with that rule placed a particular premium on the ability to face death with stoicism and dignity which is something that only men were perceived to be capable of doing.
The perceived ability of civilized British men to face death without fear was very literally interpreted as proof of their superiority, not only over women, but over men of weaker and less emotionally reserved people. You're not seeing the "real" social structure of that society any more than you would be if you examined the personal histories of Japanese kamikaze pilots, what you're seeing is a culture with a bizarre and warped attitude to emotion, to dying, and to the value of dying.
If you want to see the real "desire to protect women" in action, consider the fates of women throughout history when encountering armies of "expendable" men during warfare.
All those men staying behind to ensure their kids and wives lived is all so that men can continue to rule and dominate women? In what way does dying secure your rulership if that is the case.
This assumes that perpetuating the individual rulership of men was the objective, when in reality it would be more accurate to say that the objective was to live and die as a man worthy
of ruling. Male superiority, and the rightness of male rule, was not a convenient excuse or deception for these people. It was a sincere, deeply held ideological belief that transcended the individual self-interest of those who believed it. You think those men on the titanic are the only people in history to ever die in order to uphold a principle?
Add to this that the vast, vast majority of those men did not individually choose, but rather the choice was made for them by a small group of highly motivated professional sailors well versed in maritime custom. Factor in the enormous impact of social class in determining who survived the sinking of the titanic, and the overall picture starts to suggest something more might be going on than some ancient, primordial desire to protect the womz.