This is like proposing to dig a bigger basement to fit more flood water. No matter how much you dig, there will always be more water in the ocean to flow in.
In the case of New York, it will never have "enough" housing - what will instead happen is, eventually, it will become so overgrown that it will be unattractive for new arrivals.
Where it's the people that attract new arrivals, you may not want to move there, but there are plenty of people who do. Look at world class cities filled with skyscrapers (or even those not), they're still growing, not shrinking.
In the case of New York, it will never have "enough" housing - what will instead happen is, eventually, it will become so overgrown that it will be unattractive for new arrivals.