Chivalry was developed in a time when women were encumbered by heavy, voluminous skirts, stiff corsets, and clumsy footwear. Most chivalrous acts were shows of affectionate, and necessary, care. But right now I and most other women in the West are wearing jeans, barely anyone wears a corset, and we're likely wearing flat shoes.
But that to me doesn't mean chivalry is useless or that we should do away with it completely. There are certain things that are no longer practical (men laying their jackets across puddles, for instance) and shouldn't be done. Other things are okay, so long as you are given permission or have had a brief exchange about it beforehand, it can be something as simple as "may I take your coat?" or "let me get your chair for you." Respecting other people's autonomy and boundaries is very important, if your attempts at being 'chivalrous' make another person feel uncomfortable or demeaned then you've missed the point of chivalry.
To a certain extent, chivalry should be adopted as just general good practice. I am just as likely to hold the door open for another woman as I am to hold it for a man, child, elder, etc. I don't take offence when a man holds the door for me, and i hope no man is insulted if i hold it for him. Certain things should only be done for women you are close to (ex. Pulling out a chair) and other things are just ridiculous and I don't understand how they ever got started, standing whenever a woman enters the room should absolutely stop, and I'm glad it mostly has.
So yes, chivalry is outdated, so we should update it. Discard and adjust certain parts and just accept others, men and women, as simply good manners.
Women have proven beyond all shadow of a doubt that they are fully equal to men in every facet. Women do not need men to hold doors for them or open car doors. Women are not damsels in distress, fearful lambs in need of a savior. They are strong and don't need chivalry.
Courtship is about mating. Sex. Females attract as many males to her as she can, then chooses. Males pursue females. While she is making up her mind, it's a good idea to put the best foot forward that he can. An attractive woman will have many men vying for her attention. This is where chivalry can show her he's a man willing to look after her, be there for the long haul etc. People say that it's outdated because women are now financially independent. But mating is about sex. Not money.
Chivalry is not a concept that can be argued in terms of equality. It is merely that some men still are gentle net and know how to show consideration to women, especially women they like/love. Women aren't helpless, it's true, but we still like to be treated like women occasionally.
The origins of Chivalry came from the days of knights, identified with honor and duty. Some of us were taught this at a young age, and we still conduct ourselves in this matter.
You cant put a price on integrity, as today we see many shy away from this and when caught up in controversy, the response is typically in the theme of "the other people did this too."
Some still find time in our lives to act with Chivalry.
Chivalry still exists. Interestingly enough, I feel that with the feminist movement, some women don't appreciate it. They'd rather feel more independent. However, plenty of men will still open doors, pull out chairs, etc. And there are women who still appreciate it. So while it may be not as important as it once was, it is certainly not outdated.