The New York Times put out an article just yesterday[1] that directly addresses most of your doubts.
In summary:
* Substantially more people have died in both NYC and NY state than is demographically expected, suggesting a significant effect from the virus.
* Virus deaths are probably still under-counted, as bodies retrieved from homes aren't being tested. Patients with significant comorbidities may also be being mis- or under-counted.
I live in NYC, and all of my local hospitals are at the limits of their normal capacities (many have been able to expand their capacities, thankfully). The city has been asking for volunteers to come out of retirement.
To my knowledge, none of my friends or family have been able to get non-urgent treatment or tests for coronavirus symptoms; the system is completely focused on (and occupied with) treating those most at need.
Edit: I also forgot to mention: I don't know exactly what you're referring to w/r/t "mass graves," but I'm guessing it has something to do with Hart's Island. The city buries its unclaimed dead there; any uptick in activity probably tracks roughly with death's among the city's general demographics. It wouldn't be surprising for them to need a few more trenches.
[1]: https://www.nytimes.com/2020/04/10/nyregion/new-york-coronav...