Being chivalrous is being kind, polite and courteous, especially towards women. I don't see anything in that requiring men to pay for everything, being the "romantic instigator" 90% of the time, not behaving the same towards others or neglecting themselves.
So, if we are talking the correct definition of chivalrous...no... it's not sexist. It's being a well mannered gentleman with integrity.

Chivalry update, it used to be the polite thing for a man to walk on the outside of a lady. This was for many reasons, including to protect her from being splashed by passing carrages etc. It is now more correct to walk on the inside . To protect her from rufians and muggers that might lurk in shop doorways or behind bushes.

I don't think basic human politness is ever considered sexist.

Not sure that I agree with the things you define as chivalry are the same as what I define as chivalry. I never heard of not letting the women pay as being chivalise, nor being the instigater of of romance, nor playing socially defined gender rolls. Holding the door for someone to go first that is just manners, I do that for anyone.
Chivalry is based on showing respect and courtesy for women. It is based in the behave at Court towards women, where women were required to wear long gowns and other restictive clothing, it included things like bowing and kissing their hand when you greeted them, pulling out there chair for them to seat, opening the carriage door and offering a hand to assist them into it, throwing down your over coat so they didn't have to soil their clothing in puddles, and yes your did hold the door wide open so they could get through it in there mass of petticoats. It also included standing up and defending women and children when they were being denigrade or threatened by another man since mens behavior and action can do great harm to both that they cannot physically prevent themselves
Some of these things need to be practiced today for some of the same reasons but, especially standing up for women and children.

erm not letting someone do something isn't right really but doing something for them is nice. not sure that's even the answer to the question but in my head, there's a difference. I just haven't got the words to explain myself.