It depends on the definition of chivalrous to which the querent refers:
For those comprising a mounted warrior class, I'd say Tibet is most likely one of the last few outposts of chivalry in general, with particular utilitarian value for transportation and the elderly (two things mentioned in the question);
For those behaving in a moralistic and courteous fashion… Well, it varies a LOT based on the observer…
- Travellers tend to find Japan, Bali, and Fiji all amazing as countrywide oases of civility and compassion, though many residents would heartily disagree (for the record, I think most people, and especially drivers in Japan are very polite, but subway etiquette is totally cut-throat with respect to giving up seats to pregnant women, people travelling with small children, invalids, and the elderly).
- Residents (who know the "rules on the ground") often mention London of all places (though the rest of the UK gets pretty low marks). In the US, opinions vary widely, with cities like Minneapolis and Chicago often winding up side to side with New York City(??!?!?) in polls and surveys.
Overall, the common theme I've found in studying this issue, as well as my own travels, is that what represents courtesy (including things like not looking directly at strangers in Tokyo or NYC by default) varies quite widely, and that your best bet for being treated well in any city is to have respect for and understanding of the distinctive ways in which it operates to keep its residents from killing each other on the streets.
Wish I could be more helpful or specific than that, but the best counter-statement I would use is this: behaving in a foreign city as you would "back home", and expecting similar treatment, is one of the surest ways to ensure you will not be treated with respect and common courtesy.
In which cities around the world are people most publicly chivalrous?