Honestly- I don’t care. I’m not particularly bothered by cultural norms that teach politeness, nor if politeness in American culture (especially the South) dictates a bit of extra niceness to women. Who knows, it could help make up for, I don’t know, the rampant sexism in our society and maybe it could help teach people that, women are people too and not pleasure objects or whatever. If it floats your boat and helps you in your pursuit of virtue have at it, and yeah, it shouldn’t be a completely blackballed term.
That being said, if you’re going to play this whole chivalry game, you respect not just certain kinds of women who are “feminine enough” to merit your graces, but everyone. That means you open the door for the models, the good Catholic girls, the tomboys, you offer your coat to the nerds, the feminists, the housewives, the athletes, you give up your seat the hippies,the i-bankers, the lesbians, the rude people- EVERYBODY. To do otherwise reinforces the really crappy ideas of “chivalry” and sexuality that came to a head post- Industrial Revolution.
(Like seriously- think of the chivalry of the Victorian era and remember- it was this lovely sexual code that made it completely okay not only to give up your carriage or lifeboat to a woman, but also to annul a marriage because the husband found out that his bride had- egads!- PUBIC HAIR!!!)
And you do not hold them to some sort of standard of femininity, or see your actions as means of teaching this poor, wayward society what proper, Catholic, biblical gender roles… No. No, and No. You are polite and kind because it is the decent and charitable thing to do.
So: tl;dr- if you want to be charitable and polite, that’s fine. If you want to reclaim the word “chivalry” to reflect a just, equitable, and respectful society that’s great. But do not use your actions to wrestle others into their “proper place”, and do not mistake western cultural practices for God’s vision for humanity.