Central Paris has a residential density higher than NYC despite composed almost entirely of 4-5 story mixed use walkups. There are lots of reasons that this is a pretty ideal, human scale existence.
Those 4-5 story walkups in the East Village have a population density of 113,000 per square mile. If a half of NYC had the same density, our population could be 16 million people which is 2x the total population in NYC.
If the greater NY metro area (13,000 sq mi) had the density of 4-5 story walkups in the East Village we could house 1.5 BILLION people or many times the entire population of the United States. Of course all of this assumes linear scaling which is entirely inaccurate but the point is that 4-5 story walkup density is entirely sufficient to meet housing needs.
The challenge is getting other neighborhoods to 4-5 story walkup equivalent density.