Well, not so much "non-binary" or "non-sexual" as it would gender-inclusive.

What I object to is the default expectation that society should "train" men/boys into conforming to one set of behavior under the pretense that it will help them get laid a lot more easily. Or that every member of the opposite sex whom they encounter is necessarily somehow a romantic prospect.

My premise is that "chivalry" should be redefined as multidirectional. People of any biological sex, gender identity, or sexual orientation can practice it. And it can be practiced toward anyone -- not just exclusively in pursuit of sex or dating/courtship -- based on what it means to be a good human.

There ought to be no prerequisite for a male/female binary attached to a monodirectional "chivalrous" relationship. If a woman and a man who have feelings for one another wish to interact under such an old-fashioned arrangement, individually, then that's their prerogative. But it shouldn't be drilled into children, in general, across the board, from such a young and impressionable age.