>Sure, there's public transport... but only until it takes six times longer than driving a car

if everyone is driving, noone is. This is simple game theory and a system fault happens when there are too many cars. You can't widen city streets.

For example: public transportation in NY is often faster and cheaper than a car + parking.

I would argue that a public transportation network is a requirement, maybe even a prerequisite, for high density. Manhattan simply could not work without the various public transportation methods -- if everyone commuting in from CT, LI, NJ, upstate NY, etc., had to drive in, would there even be enough space for all those cars on the island?