A "progressive" city can still have legacy parts, and those legacy parts are supplemented by custom features. All regular NYC Buses are handicap accessible. There are 24/7, door-to-door, paratransit services for handicapped people that cost the same as a subway fare [0]. Sure, the NYC Subway isn't Seoul or Singapore's, but it runs 24/7 (Seoul's only runs 5:30-Midnight) and the stations on average are probably ~20-30 years older than the oldest parts of the Seoul Metro (early 70s). The unofficial handicapped accessible subway map is depressingly sparse [1], but it seems on par with London's [2] and better than Paris' [3]. (Neither of which are 24 hour systems, BTW.)
I'll admit that the surfaces are gross and the subway collects the most unfortunate, down on their luck members of society, but your post reads as a total condemnation of the best mass transit system in the US. It's dirty and depressing at times, but it's also one of the great equalizers in New York City life.
[0] http://web.mta.info/nyct/paratran/guide.htm#what_is
[1] https://subwayrecord.files.wordpress.com/2015/03/ada-subway-...
[2] http://content.tfl.gov.uk/step-free-tube-guide-map.pdf
[3] http://parisbytrain.com/wp-content/uploads/2008/09/plan_pbs....