Great NY style pizza is about:
50% bake time
23% skin thickness (aka thickness factor)
10% proper fermentation
7% other (formula, launching skills, cheese, etc.)
Bake time is more than just the temp of the stone. It's a combination of the temperature of the stone, the thickness, and the material the stone is made out of- and the conductivity of that material. You can have a material that transfers heat slowly that will bake a pizza in 10 minutes at 550, while you can have another material that transfers heat quickly that bakes a pizza in 3 minutes at 550. Temp is a part of the fast bake time equation, but it's not the entire equation.
Home ovens temps usually straddle a line whereby, if you use a poorly conductive stone, such as a typical baking stone that you might find in stores, you'll get long bake times and, imo, usually mediocre pizza, but if you use a material with higher conductivity, such as steel, you'll get shorter bake times, and, with those shorter bake times, better oven spring.
The thickness of the dough isn't as important as bake time, but it's still pretty important. The necessary skills to stretch a dough as thin as it should be for NY don't occur overnight. It takes considerable time to master stretching- it also requires major fermentation knowledge to achieve a dough that's easy to stretch.