It's not about supply by itself, it's supply vs demand.
Big cities have big populations and economies, which means a lot of demand.
But as far as zoning goes, note that like other metros, NYC has actually effectively downzoned areas over time; in fact, a huge proportion of buildings there would now be illegal due to different types of density restrictions: https://www.nytimes.com/interactive/2016/05/19/upshot/forty-...
A lot of NIMBY's tend to go, "well, there's still some areas where it's possible to build more, so there's no real issue there". This is approximately like restricting car manufacturing to one state, and then when car prices go through the roof, saying "well, there's still more land to build car factories there, so I don't see the problem, it couldn't possibly be this rule".