Chivalry is about sex. It will never die because it is about getting laid.
A man who pays for dinner or opens a door for a lady is demonstrating his ability to provide and showing that he values her. But at the bottom of it, he’s doing it because it will get him laid. He knows this, and she probably knows it too.
This goes back to when caveman would share meat from an animal he’s hunted with a female in order to gain favor with her. (We’ve observed the same behavior in chimpanzees.) Clearly, a male who is skilled enough to hunt—and tough enough to keep his catch from other males who would steal it—is one to consider as a mate.
Interesting: a male who is skilled enough to hunt but lacks the social courage to stand up for himself and stop a more aggressive (though less skilled) male from stealing his quarry will not have any meat to share with the female, and thus be of no use to her. The asshole male, however, has the meat to share because he is an asshole. This may be an illustration of why females respond more positively to a male’s self-confidence and assertiveness than his skill set.
Sex is a reward that females give to males who turn them on, who perform well for her in some way or another. Withholding sex because of her mate’s inability to perform well is a practice unique to females. (What’s remarkable is that she does this even though the female enjoys sex herself.) Females recognize instinctually that sex is their strongest leverage over males—they have us by the balls at all times. A suburban wife won’t have sex with her husband until he gets off his ass and paints the house; a cavewoman wouldn’t mate with a male who used to be a good hunter but now comes home empty-handed. Women are still at work selectively breeding out the lazy and incompetent males from our species.
A male withholding sex from a female just doesn’t have the same motivational effect.
Thus the man is expected to drive, to hold the door open, to pick up the bill—if he wants to get laid. Nature demands chivalry, and that will never change so long as there are humans.