The offering of chivalry and the notion of equal rights for women can co-exist quite easily. The dividing line becomes something that’s more nuanced. It’s a recognition that there are differences in how people get treated in a professional/ workplace environment and how they are treated when in a social situation.
When it comes to the workplace, women and men have the same rights to compete for jobs, to receive fair treatment, to get paid the same if they do the same work in the same circumstances etc. Both men and women benefit when companies hire the best qualified people to do whatever needs to be done. I also would say that there is a space for what would be common professional courtesy in a workplace whereas gestures of chivalry would not be apropos.
Equality in the workplace does not mean that in social situations men and women are the same. It seems that when the issue of women’s rights got elevated during the late 1960s and early 1970s it became conflated with an idea of “sameness.” But as social beings the genders are far from being the same. There are many ways in which our different tendencies and ways of thinking are now more clearly understood and accepted. It’s now clear that we have some different chemical reactions going on in our bodies when the same type of stimuli are encountered.
For example, I heard John Gray speak to a group of men and women and talk about what a man should do when he hears his wife in the car (or girlfriend if he’s not married) say something to the effect of “I’m getting hungry.” He told the men that their primary mission was to find the first place that had food available and stop there. It got some laughter but a recognition that it was true. I have heard speeches talking about the differences in selling to men and women because of how each goes about the process of deciding whether to buy something (speeches presented by women and by men). We're not simply the same creature with slightly different body parts.
Socially there are some differences in the roles among men and women. Of course chivalry isn't mandatory, and if a couple chooses something different, they're certainly free to interact however works for them. In fact, one of the pillars of feminism is for a woman to be able to choose that -- so why would people deny women the option to welcome and accept it? For the large majority of women, chivalry is still a desirable trait to see in men. For men, it gives them a bit of a roadmap of behavior after a period when it seemed that all patterns were tossed out the window and guys were scrambling to figure out what was expected of them.
Younger men and boys face a challenge of knowing how to treat women with respect in social situations in particular. You don’t hear rap lyrics about women not deserving to be paid the same as a man, nor her right to be a CEO. But many negative messages about women socially come through in too many songs, whether the words used to describe women or talking about sexual acts in perverse and demeaning ways. It’s on the social side that better role models are perhaps more important than ever.