I think you may be articulating the very point I am trying to, indirectly.
While gender as a rigid dichotomy is something I would agree is an anachronism unfit for modern society, I would argue further that conduct that could be described as chivalrous is by no stretch, or at least not necessarily, an immediate jump to viewing people based solely on gender and nothing else. You mention the term 'fairer sex,' and there I find a most excellent example. In that phrase you will find nothing condescending, and offense only with those who are looking quite actively for something to be angry at. Indeed, were you to ask me I would most candidly reply that not every woman out there could be even remotely described as fair. Or if you'd like to put it another way, not every woman is a lady.
But one asks the question to find little in the way of response (and here I do not include you, specifically) save for a deal of "I treat everyone EQUAL! Everyone is the SAME! See how modern I am!" This, I would propose, misses the point entirely. At least with regard to such gestures as the door or chair maneuver, such gestures carry no more sexism than gender-specific pronouns.
And of course, let it be said that he who holds the door for a comely young lady and leaves it closed for an elderly man or harried parent with baggage to match is no gentleman. Chivalry, as many in this thread seem tireless in failing to notice, is more than such flattering acts as may be hoped to help secure a lady's favor.