We don't have chivalry because we don't need chivalry. The demand for chivalry has taken a nose dive. There are more than one reason for this, but a big one is technology.
Technology has made everyone richer; by leveraging the power of technology, everyone has more power over their environment, more control of their personal circumstances. This cheapens the value of chivalry. Who needs a man to hold the door when a computer opens the door for you.
If there is a real war with a real enemy at our gates, all this will be swept away in a flash. The women in the army will melt away, the homosexuals in the army will stay if they can contribute physically and mentally to killing the enemy, but their behavior patterns will not be tolerated. The instinct for species survival will kick in, and homosexuals have nothing to offer in this domain.
Women and homosexuals in the military are just two out of many manifestations of our very wealthy society. When we can no longer afford them, they will vanish like a morning fog. For now, we can afford them.
You make a good point. Our great wealth, ubiquitous conveniences, and relative national security have afforded us the luxury of living in a comfortable world that has enabled us to engage in all manner of social experimentation. However, this can all be quickly swept away when an existential threat arises. Then it will be back to reality.