Unfortunately NY is set up as a centralized archipelago; existing transport infrastructure is pretty much saturated, and encouraging outlying areas to gain jobs is not workable because Manhattan is the only place accessible to every single part of the metro area within one or two seats on public transport.
It's hard to use the tax boost to add more mass transit, because you'd have to bond based on projected revenues, which is fickle at best. The only places that tie redevelopment and transit together successfully seem to be in Hong Kong and Japan, but in those places the mass transit operator either already owned the land being redeveloped, or got it for pennies on the dollar as a de facto public subsidy, which would be impossible in the US due to the Just Compensation clause of the Fifth Amendment.
That's why I looked up how much of the city is residential right now. If you only spend 1/3 of the city on apartments, that leaves you a lot of room for everything else.
> What about playgrounds and parks?
Oops. I had a thing about making an entire block into a park out of every 3x3 but I lost it in editing. So yes there's lots of room for parks.
> And can all these workers get to work using the available infrastructure?
The base idea was set in 1900 and housing the same number of people but in better buildings, so yeah they can still get to work.
For the idea of fitting more people into the current city, I'm not sure, but it would probably work out. Have more mid-rise housing right next to jobs, and use the tax boost from having more residents to add more mass transit.
As far as zoning, the issue is not housing vs. services and infrastructure, it's the size of the housing. Detached houses have an awful use ratio.