I believe you stated the issues correctly.
A "true" or "authentic" NY style as made by professional pizza operators is usually (almost always?) baked on some kind of stone surface. Consequently, the amount of sugar in the dough is kept at fairly low levels, or omitted altogether, so as not to lead to premature browning or burning of the bottom crust. Some operators who like to include some sugar in the dough for flavor and other purposes (e.g., crust color) will sometimes put the unbaked pizza on a screen and put the screen directly on the stone surface. The screen in this case keeps the pizza a slight distance from the baking surface so that it doesn't overcook and burn. To get a better bottom crust (e.g., a crispier crust), the operator can slide the pizza off of the screen onto the stone baking surface toward the end of the bake and let it finish baking there. Commercial deck ovens have very large stone surfaces, so pizza size is usually not a limitation. They can easily handle, say, 16"-18" pizzas, and even larger, without a problem. In a typical home oven setting, size is usually an issue. The largest size pizza I can make directly on my pizza stone is 14", and just barely. For anything larger, up to 18" (the largest size my oven can accommodate and be able to close the oven door), I use a screen, as addressed in my last post.
Most chains like those you mentioned use conveyor ovens. So they will use either screens or disks (and sometimes pans). The American style pizza made by the chains usually contains a fair amount of sugar in the dough. In most cases such a dough would brown prematurely or excessively if baked directly on a hot stone surface as used in a typical commercial deck oven. Conveyors operate at lower temperatures than deck ovens and the pizzas are usually cooked by heated air so the bottoms of the pizza crusts don't burn or brown prematurely. It is possible to bake an American style pizza on a stone surface but you would want to keep the sugar levels low or else use a screen/stone combination as mentioned above. This is less a problem in a home setting but even then you may want to experiment a bit with sugar levels until you get a better feel for what results you can get using your oven.