I think you're both sort of right? NYC is dense for a US city, but not really so for a global city. We're sort of space constrained on an island like Manhattan -- since it's an island -- but much of the outer boroughs are 3 story buildings (with of course exceptions in places like downtown brooklyn, williamsburg, LIC, etc.) There are places in Manhattan we could build more high rise condos, but it's probably easier to focus on the outer boroughs. No one is going to bulldoze brownstone brooklyn, but if we even built more like Paris, which has a population density of almost 2x NYC without skyscraper condos, there certainly would be a huge amount of extra capacity.

I'm not a big fan of what happened to my neighborhood (Williamsburg) with all the giant condo buildings that have gone up over the past decade or so, but people do need places to live and we probably should be growing denser across the entire city. There's definitely a lot of space left, just not a lot of will to change things.