Chivalry is the quality of a man who can tame a horse and ride it to war.
That's all it is. Everything else associated with it is either a precondition for developing this quality, or a consequence of it. The core of the thing is the man and the beast.
What does it take to tame a horse? It takes courage, not recklessness, but that kind of disciplined and developed courage that comes from learning to fear being thrown, and getting on horses again. It takes self-mastery, because the horse is a prey animal that will amplify your fear. You must learn to ride through it, until even you don't really feel the fear in the same way anymore.
It takes gentleness. A horse responds to the slightest touch. You must be sensitive to its movements, its breathing, the language of its body.
What does it take to ride a horse to war? It takes trustworthiness. The horse must believe in you to charge into the smell of blood.
It takes honor. You can't ride alone. You must build relationships with other men like you, who know they can count on you while there is blood in your body. There is your self-sacrifice, even to death.
What does it build in you to do these things? Some of the things have been said. You get the virtues you practice, as Aristotle teaches in the Nicomachean Ethics. You must have some courage to begin, but you will build courage as you do. You must have some self-mastery, but you will become the master of yourself. You must be gentle, and able to understand another very different kind of living being through touch alone. You will become moreso.
The habit of keeping your word is like any other habit. After a while, it becomes part of you. The habit of honor likewise.
Can you do without chivalry? I don't know. Can you do without men like this?
That's what chivalry is, and that's the root of what people come to call an 'ethos' or a 'system.' But it's easy to be fooled by the accidents. What US Special Forces were doing in Afghanistan a decade ago was not different from what Charlemagne's riders were doing in this core way. It is different in other ways, accidental ways, but this is the essence of the thing.
You can see, with a little thought, how the thing we call courtesy grows naturally out of this combination of gentleness and trustworthiness, a habit of honor and a character that masters itself.
The question to ask is not whether you need chivalry, nor even whether or how to revive it where it is lost. The question you should worry about is whether you can do it without the horse. I think you can, but only if you can recapture that essence in other ways. Wherever possible, it is best to start with the horse.