You can have chivalry or equality. You can't have both. What everyone can have instead is politeness, compassion and/or humanity. Chivalric behaviour automatically assumes a power imbalance that has to be tacitly accepted by both sides... in the case of you and the lady with the cane, the imbalance was one of physical ability - your act could be considered chivalric, though I would've preferred the term 'kind' and/or 'compassionate'.

I'd question Ms Jgln's assertion that 'Women and Children First...' rule doesn't exist. I'll go and read the article in a second, but all we have to do is look at the events in Ukraine to see the rule is tacitly accepted by populations as a whole. Men and women are both equally capable of firing javelins into oncoming tanks or throwing molotov cocktails into jeeps - yet only the men have been conscripted (though women can opt in if they wish). We don't question the conscription because we have internalised this rule as sound and sensical - and it'd be a brave feminist to stand up and insist on equal treatment at the moment.