I live in NYC too, and we actually have a lot of resources for the homeless. New York City Department of Homeless Services (DHS) is pretty robust with a lot of programs and solutions for homeless people with different needs.

You can even call 311 to let them know if there's a person living on the street and they will come help get them connected with resources to assist them.

Extreme cases of homeless people with mental illnesses are the most visible and disruptive, but they're actually a very small percentage of the homelessness problem. But there are resources for them, they're just also the most difficult to reach.

I'm by no means an expert on DHS, but look into them they're doing pretty great work.