Did you know that the first time a moving sidewalk was proposed for New York City it was 1871? It was brought up again in 1902 for the Brooklyn Bridge, according to EphemeralNY, after which it was debated in the newspapers. Eventually Mayor Seth Low spiked the idea, forcing generations of pedestrians to use their own two legs to get anywhere. Until 1910, when the idea rose from the dead! This time, they were to replace the subway system... which sounds like a highly unenjoyable, fume-heavy experience.
To backtrack, in 1871, the idea came from local wine merchant Alfred Speer, who patented the first "endless-travelling sidewalk"—his idea was to build an elevated one moving along Broadway. It was suggested that boring "stop 'n' chats" would become a thing of the past as one would just have to "step on the passing sidewalk to be carried rapidly beyond sight or hearing of his tormentor." Cause of death on this one: no financial backer.
New Scientist reports that by 1902 New Yorkers were fed up with the rush hour crush; one commuter calling it a "daily torture." So Bridge Commissioner Gustav Lindenthal figured some high-speed moving walkways oughta do the trick, and proposed a looping system with 4 walkways on the Brooklyn Bridge, the fastest containing benches that would whisk pedestrians across the river. It's suspected that Brooklyn Rapid Transit had a hand in burying the idea, as they had a monopoly on the borough's public transit at the time.
Still, proposals came in, with moving sidewalks suggested for the Williamsburg Bridge, Wall Street, and as mentioned, in the subway system. The New York Times even wondered "why this improvement was not considered when this present [subway] system... was built". And in 1932 a similar idea involving an elevated tube system was proposed.
Take a look at the moving sidewalk in action, in Paris in 1900. And if you think the idea of incorporating those into the city is crazy, check out what the world thought 2008 would look like back in 1968.