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.