Can you, though? Any given neighborhood has limited space to build things. At some point, you have to start building high-rises all over to pack more people in. This also has a limit, so beyond that you'd have to start building over parks etc.
This argument is overblown to the point of absurdity. Hong Kong, a city known for its high rises, has 1/2 the density of Brooklyn, a place which is hardly known at all for its high rises, and 1/3 as dense as Paris, which has limited height city-wide by statute. Neither has a lack of parks or public spaces.
You don't need high rises to build more homes. Seattle, at Brooklyn's level of density characterized by mid rises and townhomes, could house 4x as many people as it does today. And it doesn't mean infrastructure and public services would suffer any more than it does for New York, London, Berlin, Paris, Tokyo, or any other dense first world city. In fact, it might even get better.