Welcome to NYC!!! Please remember that NYC is huge and you can never go to "just the right place" – besides that "it" often changes so be sure a place that is recommended is still open,
As a Noo Yawka, I think the "NYC" experience is just being here, seeing whatever and whoever, and being open to eating whatever too. I like the idea of street carts becuase they are "real" and you don’t waste 1 or more hours of time sitting inside when so much of the NY action is outside! Just walk on the streets, go to the parks, find out some free or inexpensive NY Sumemr entertainment (concerts, etc. in parks) and enjoy! and come back for more (then you can give us YOUR NYC roadfood discoveries!
Everything you wrote is absolutely true. I think of NYC as a walking town and I try to avoid the subways as the action is above ground.
My favorite thing to do is to walk by every restaurant and read the menus posted outside.
With this in mind and considering most visitors will go up 42nd St. to Times Square, I recommend that visitors walk a block or two more and hit 9th Ave. then turn right and head to Central Park. This is where your suggestions of free entertainment and people watching hits its zenith. But on 9th Ave, there are plenty of great places to eat. Try Island Burgers or one of the many Thai places or…..