From what I've gathered from news and 60 Minutes coverage, although they didn't mention mass transit nor elevators (which is probably part of it) and the very high population density, yes, hospitals were overwhelmed.
Also, certain poor communities were hit very badly (e.g. the Bronx). People of color were hit more badly. These poorer communities tended to have more people with pre-existing conditions.
Poverty, pollution and neglect: How the Bronx became a coronavirus 'formula for disaster'
The Bronx's plight is just one example of how the virus has disproportionately impacted minority communities around the city and nation. Hispanics accounted for 34% of COVID-19 deaths in New York City though they make up 29% of the population, while African Americans made up 28% of the deaths, but 22% of the population, according to data released by the city. Early data from the Centers for Disease Control and Prevention also shows African Americans are disproportionately shouldering the impacts of the virus nationwide.
Home to the poorest congressional district in the country and what researchers say is the unhealthiest county in the state of New York, the Bronx has long been plagued by underlying illnesses now associated with an increased vulnerability to COVID-19, such as respiratory problems, diabetes and heart disease. More than 91% of Bronx residents who died from COVID-19 had underlying conditions as of April 10, according to New York City Health Department data.
For decades, the borough suffered under some of the worst air pollution in New York City. The Bronx contains seven of the city's top-10 neighborhoods for asthma rates, according to an analysis by housing-data website Localize.city. The Morrissiana/Highbridge neighborhood, home to the city's second-highest asthma rate, is now among the city zip codes worst hit by the coronavirus, according to health department data.
As COVID-19 deaths continue to rise in the Bronx, Johnson said many residents feared their historically polluted air had left them particularly vulnerable to the virus.
There's a lot more. Search the above for health.
I probably watched Sick doctors, nurses and not enough equipment: NYC health care workers on the fight against the coronavirus when it aired. I know I watched "Beyond anything I've seen in my career": Doctors on the front lines describe surge in coronavirus patients when it aired.
And, Gov Cuomo has said that (over there), 80% of those who end up on a ventilator from COVID-19 die.
Although not directly mentioned in the news stories I've seen, for infection, you've got at least 3 airports that common hubs/stopping points for travel: EWR, LGA and JFK w/tons of people passing to/from Europe besides NYC being a tourist destination. Notice how badly much of Europe has been hit per COVID-19 Map
I wouldn't be surprised if part of the reason for the decline in NY deaths is for similar reasons to NPR Choice page.