Enforcement on car inspections is lax. You might receive additional citations or an increased fine for having an expired inspection, but a lot of New York drivers (who I assume are not unique) will simply ignore tickets or pretend that they never received them (which is easy when they frequently come in the form of a letter tucked under your wiper, rather than a digital record).

There are plenty of cars in my neighborhood that are patently unsafe by the city and state's standards, or are modified in ways that wouldn't pass inspection. Many of them belong to cops, who drive into my neighborhood to police it from outside of the city.