We have several members who routinely use only flour, water, yeast and salt for their pizza doughs, including for the NY style. In fact, if you study the history of the evolution of the New York style, you will find that the early doughs included only flour, water, yeast and salt. These were the ingredients used to make the Neapolitan pizzas from which the New York style evolved. The pizzas made from such doughs were baked principally in very high temperature coal-fired ovens, which corresponded to the very high temperature wood-fired ovens used in Naples.
It wasn't until the invention and commercialization of the deck oven, along with lower bake temperatures and longer bake times, that oil and sugar started to be added to the doughs. But the amounts of oil and sugar added were not large, maybe 1-3% oil and about 1-2% sugar. At those levels, you aren't going to get a lot of tenderness in the finished crust. In the case of the oil, you will get some coating of the gluten strands and an increase in plasticity and extensibility of the dough, and also some flavor, but not a great deal of tenderness. You would need above about 4-5% to get that effect. In the case of sugar, you can't use too much because the bottoms of the crusts can brown prematurely or even burn. When refrigeration was commercialized so that pizza operators could cold ferment their doughs if they wanted, the sugar also helped feed the yeast over long cold fermentation periods and also contributed to final crust coloration. Sugar, especially when combined with oil, can also provide crust tenderness, but, again, you will need a fair amount of it, much more than just a percent or two. High oil and sugar levels are more commonly associated with the American style of pizza (think Papa John's) than the NY style.
I read somewhere that there are close to 1500 restaurants in NYC that include the word "pizza' or "pizzeria" in their names. With that many pizza places, and given that a lot of pizza operators are unfamiliar with the intricacies of dough, there will no doubt be some who use oil and sugar outside of the ranges mentioned above. Some will even include ingredients like milk and eggs in their NY style doughs. Some, like Sbarro, with which you are familiar, have used nondiastatic malt in lieu of sugar, and some might even use honey. When pizza operators are unconstrained by knowledge, just about anything can find its way into a pizza dough and in quantities that can range all over the place.
With the evolution of the NY style over a period of decades, "purism" can mean different things to different people. If you grew up in the early days of the old masters, purism would have meant using only flour, water, salt and yeast and using a very high temperature oven and short bake times. And the yeast would have been fresh yeast, since dry yeast hadn't yet been invented. Today, it might mean using flour, water, salt, yeast, oil, and maybe sugar, and lower bake temperatures and longer bake times. And the yeast might be dry yeast, which has increasingly supplanted fresh yeast for pizza dough (although there are still many pizza operators who continue to use fresh yeast). Some may consider it heresy, but ten years from now the dominant oven, not only for the NY style but for most styles, might be the conveyor oven and using screens and disks and pans in lieu of stone baking surfaces. You would be surprised how many pizza operators have switched from deck ovens to conveyor ovens, and not only the major chains.
If your are a history buff, and you haven't already done so, you might want to read the thread at http://www.pizzamaking.com/forum/index.php/topic,14920.0.html
. That thread makes for a fascinating read and will give you many insights into how the NY style pizza as we now know it evolved from its humble beginnings at the start of the 20th century.