Part of it is household size; the modern household has decreased in size dramatically, now that raising children has moved from "we need as many children as possible so that at least a few of them make it to adulthood" to "giving children a quality life is expensive".
A good deal of it is transportation. In 1800 most people walked, and the rich used horses. By 1900 we had railroads and streetcars; by 1920 we had metros; by 1950 we had automobiles. The effect of the automobile has been strongest in America; a single highway interchange in Atlanta takes up as much room as the city of Florence, Italy. https://www.treehugger.com/urban-design/you-cant-set-shop-si...
That section eyeballs to be around roughly ~1.5 mi across. Larger than 2000 ft but a lot less than 5.5 mi.
Living standards affect density, but they're not the reason we can't fit 190 people per acre.
Let's say you're in 1900 making 6 story apartment blocks. You could set up a residential district that's 30% road, 20% open space, and 50% building.
Make your average apartment be 600 square feet with a 20x12 living room, two 10x12 bedrooms, a 10x6 kitchenette, a 10x6 bathroom, and an occupancy of 3. Probably slightly reshaped for the sake of windows.
With some extra space for hallways and stairs, you can reach an average ground impact of 75 square feet per person.
That means it only takes a third of an acre to house 190 people, leaving the other 2/3 for businesses. For comparison NYC is currently 75% residential.
Such an apartment isn't huge but it's still twice the size of college dorm.
These days you could build 3x as high and give each person over twice that much space and have lots of elevators and still fit 200 people into a third of an acre. Rent out the bottom floor to shops too.
Until 1961, New York had a zoning code that was concerned with exactly none of those things, and planning for services and basic infrastructure was haphazard at best. While the current zoning code is overly restrictive, it at least prevents growth in places that would not have the services required to handle it. (It also prevents the growth in a lot of other places as well, but that's beside the point.)
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.
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.
Manhattan has a density of 72,000 people per square mile. If it were its own city, that would make it the 6th densest city in the world[0]. The top three are Manila and Mumbai, at 108,000, 74,000, and 74,000 people per square mile, respectively. So, regardless of levels of development, modern cities cap out under 80,000 people per square mile, with Manila being a slight outlier (but not by much).
190 people/acre - what you cite for Antioch - is 121,600 people per square mile. I'm not able to find any reputable sources that back that statistic, and estimating populations of cities from antiquity is inherently a bit tricky. 500,000 people would also put it at 50-80% of the population of Rome during the same time period, and more populous that Constantinople, though it's not included in any of the four lists of populations of large cities on this page: https://en.wikipedia.org/wiki/List_of_largest_cities_through...
[0] There are two more, at 108K and 80K, but they both have populations under 70,000, so they're not comparable.