Density doesn't mean making spaces smaller, it just means making more of them. Usually this is accomplished with more floors. If you look at NYC, a huge portion of it is buildings are <= 12 stories. NYC is unique in its density and is certainly a model for other cities in the US, but in terms of fulfilling its own livability needs.. it will have to become denser through height. Even NYC must address its demand with additional supply within Manhattan -- up is the only way to go.
Instead what we see today in NYC is the same as what we're seeing around North America. NIMBYism, heritage protections, gatekeeping, lawsuits, FUD, political grandstanding, etc.