Chivalry is not out of style. I both see and practice it every day.
What is out of style is the old-fashioned notion that women are delicate creatures who survive thanks solely to the grace of manly men. And that should be out of style.
When I hold open a door or give up my seat for someone needing to sit, I do so because of basic human decency. It has nothing to do with gender. Men have held doors open for me lots of times, and I for them. Same with women. In the latter case, I never let fear of a feminist backlash stop me. (By the way, is that really a thing? I've never encountered a woman who got mad that I held open a door for her, and at least a few were feminists. I'm guessing that's an anecdotal leftover from the 1970s.)
People should always strive to be friendlier, or at least cordial, insofar as functioning in a crowd goes. A truly decent person is going to be as accommodating as possible, particularly in a one-on-one situation. Hopefully, that comes naturally. If not, it can be learned.
But it should never be rooted in sexist thoughts.