Thanks for the information.
I think the reason the dough is too tough to work with is because you are not using enough water. Using 8.625 oz. of water yields a hydration of 53.9% (8.625/16 = 53.9). A typical NY style uses around 60%, give or take a few percent. I typically use 63%. At 63%, you would need 10.08 oz. of water.
You oil is also too high for a NY style. If you are using 1.625 oz. of oil, that comes to about 9 7/8 t. (about 3 1/4 T.), or 10.1%. A rate of around 1-4% is more typical for a NY style. So, for now I would suggest reducing the oil to around 3%, or 0.48 oz. I would also recommend that you not mix the oil with the water. Once the dough has formed in your processor bowl, the oil can then be added. Adding the oil last prevents the oil from interfering with the hydration of the flour by the water.
It sounds like you heated all of the water to 110 degrees F and then added the ADY to proof it. I would like to suggest that you use only a small amount of the total water, say, 1/4 c., and use that to proof the ADY. The rest of the water should remain cool. In fact, I would use very cold water for that portion since your food processor will add a lot of heat to the dough in the form of frictional heat. With all the yeast you are using (one packet), the frictional heat will cause the dough to rise very very quickly. So, you want to keep the dough on the cool side, and especially if you plan to try a cold fermentation. If you decide on cold fermentation, I would cut back on the yeast by at least a half. Otherwise it will be hard to cool down the dough and it will balloon up in the refrigerator. I would also go directly to the refrigerator after you have finished hand kneading the dough. You should divide the dough into two pieces, lightly coat them with a bit of oil, and put them into covered containers to go into the refrigerator.
The dough will be usable after about 24 hours and up to 72 hours. When you are ready to make the pizzas, bring the dough balls to room temperature and allow them to warm up for about 1 1/2 to 2 hours. If your kitchen is on the cool side, it might take a bit longer.
I estimate that your formulation will produce a total dough weight of around 27 ounces, or around 13.5 oz. for each of two dough balls. A weight of 13.5 oz. used for a NY style pizza should allow you to make a roughly 13" pizza.
The rest of your procedure seems OK. I think if you follow the above recommendations and suggestions, you should see improved performance. If not, come back and let us know what you experienced. Good luck.