I think you framed the issue quite nicely. So, the question becomes how do you use the ingredients available to you and your bread machine (or food processor) and your particular oven to achieve a pizzeria-quality NY style pizza? In Jeff Varasano's case, to which tdeane (Terry) referred, he apparently found it necessary to use a DLX mixer with a special "wet knead" process and multiple rest periods, and a home oven that was modified to defeat the self-clean feature so as to produce bake temperatures in excess of 800 degrees F. In your situation, unless your are prepared to do something along the same lines, you may find that you will have to spend more time on your dough formulation than anything else to see if there is any way of adapting it to your particular pizza making equipment, specifically, to your mixing equipment and oven, to achieve results that will satisfy you.
I agree with Terry that the type of flour used should not be an impediment to what you want to do, especially for a super-thin crust along the lines you wish to achieve. However, in my own case, with my combination of a basic C-hook KitchenAid stand mixer and a basic unmodified electric home oven capable of delivering a maximum oven temperature of around 500-525 degrees F, I found that I got the best results using high-gluten flour or bread flour. If I had to guess, I would say that most NY places making the style of pizza you are after are using bromated flours. Although I have studiously avoided bromated flours, if my goal was to try to replicate an authentic NY style pizza along the lines your are pursuing, I think I would use a bromated flour. In my case, I would perhaps use a bromated bread flour or high-gluten flour. A high-gluten flour would have the added advantage of achieving a stronger gluten structure better capable of retaining the gases of fermentation, and would also add a bit more flavor and chewiness to the crust because of its higher protein content.
There is no doubt in my mind that a bread maker and a food processor can produce very high quality doughs, in many respects better than my KitchenAid stand mixer. I have tried all of the methods--stand mixer, food processor, bread maker, and hand-kneading--and they all work. However, if the predicate to making a proper dough for making pizzas is that the dough be slightly underkneaded, and if it is also an objective to minimize oxidation of the dough and harm to carotenoids that contribute to crumb color and taste, I think that using a stand mixer gives the greatest control. I discussed some of the control issues surrounding the use of a food processor to make NY style dough at http://www.pizzamaking.com/forum/index.php/topic,2189.msg19291.html#msg19291
and the use of a bread maker to make NY style dough at Reply 51 at http://www.pizzamaking.com/forum/index.php/topic,576.msg5486.html#msg5486
and at Reply 260 at http://www.pizzamaking.com/forum/index.php/topic,576.msg17113.html#msg17113
. I also found that it was possible to hand knead a dough using high-gluten flour if the hydration were made high enough and rest periods were used during the hand kneading process. However, even if the mixing issues are resolved, however that is done, there still remains the issue of getting enough heat out of a standard home oven without altering it in some fashion to get higher bake temperatures.
Against the above backdrop, when I look at my own situation, that means that I would have to look at possible dough formulations to produce the desired results. I have used autolyse and I have used natural starters and many types of preferments, both natural and those based on using commercial yeast or combinations of natural starters and commercial yeast. I personally found that autolyse produced a too-breadlike crumb for my taste but that starters and preferments produced high quality results in terms of crust flavor, color and crumb texture. But, with my oven, I am not sure that those applications would allow me to produce pizzas like you want to make. Moreover, before heading in any of those directions, and since it is highly doubtful that the typical NY style pizza operator is using these elegant solutions, I think that I would at least take a shot at a more direct approach.
So, I think that if I were to experiment with a NY pizzeria-style dough formulation that might work in my home oven using a pizza stone preheated to the oven's maximum temperature, I would perhaps use a bromated high-protein flour (bread flour or high-gluten flour), with a high hydration (maybe between 60-65%), a thickness factor of around 0.065-0.07 as a starting value, an amount of dough to make a 14" pizza (the largest my pizza stone can accommodate) and shaped to have a very small rim, a fair amount of oil to achieve softness and flexibility in the finished crust (possibly around 3%), and shoot for a short bake time so that the dough doesn't dry out and become cracker-like and/or develop too much crust color. I would sift the flour to improve its hydration, use no sugar, and salt would be at normal levels, around 1.75% for my palate. If I elected to use a non-bromated flour, I would perhaps use a pinch of ascorbic acid (Vitamin C) in the dough for dough strengthening purposes. I would use IDY at about 0.50-0.60% and a one-day cold fermentation. Under the above conditions, I would expect the dough to be fairly extensible. It is also possible that I would use some of the methods I used at http://www.pizzamaking.com/forum/index.php/topic,3985.0.html
, particularly the use of a combination of whisk, flat beater and C-hook attachments, in order to produce a better quality dough out of my KitchenAid stand mixer.
I have no idea if or how well the above dough formulation and protocol would work. Usually, it takes a few tries to prove out the process or to refute it.