It's the other way around, Tokyo has higher population density:
New York City: Land area: 784 km2 Population density: 10,431 ppl / km2
The Special Wards of Tokyo: Land area: 619 km2 Population density: 15,146 ppl / km2
Manhattan: 27,799/km2 and 59.13 km2
So that's denser than the overall special wards area, but also vastly smaller in both land and total population. Basically 1.6 million people vs 9.4 million people.
Probably significantly easier to have an extensive train network when you're consistently dense for that large an area.
But I would agree that the main reason the US's public transit infrastructure is lacking is just that we like our cars too much. I expect this to get even more pro-car in a self-driving-car world, too.
It's not just unfair, it's completely absurd. Tokyo sprawls like crazy, and a big percentage of the population lives in the outer wards. The special wards are high density, but expensive.
As soon as you go outside of the Yamanote line, the population density of Tokyo falls dramatically (probably to Brooklyn levels). Take the train 10-15 minutes outside of that, and you find lots of detached houses and open land. Yet it's all "Tokyo", and this land area is counted when you see the huge population of the city referenced online.
Tokyo has become a fantasy for people in America who think that urban density solves all problems. Most of these people have not ventured outside of the tourist regions of the city.
American public transportation systems, even here in the densest part of the country, are easy political whipping boys. Blame whatever you'd like (cars, highways, unions, politicians, etc), it appears that Americans don't care too much about fixing their transportation infrastructure if those fixes don't involve cars.