I don't make grilled pizzas so I won't be able to help you with that facet of your pizza making. However, if you use your home oven, you should use a pizza stone that you have placed on the lowest or middle oven rack position and preheated for about an hour at your oven's highest temperature. Using a sheet tray with nonstick foil and an oven temperature of 400 degrees F won't give you adequate oven spring, and the pizza is likely to bake out too dry. It is also likely to be quite flat.
I don't know how you measure out your flour, that is, with a heavy or light hand, but I estimate that the recipe you posted makes about two pounds of dough. That amount of dough can be difficult to knead by hand, especially if it is a high-gluten flour or a bread flour. I don't usually hand knead two pounds of dough at a time, but I would guess that it should take about 15 minutes of hand kneading after combining all of the ingredients to properly develop the gluten. It might take even longer if you don't have good hand/arm strength or you are using a high-gluten flour. You didn't indicate what size pizzas you have been making, but I would divide the bulk dough into the desired number of dough balls before letting them ferment.
Your recipe won't replicate a pizzeria's pizza because pizza operators don't make one-hour doughs. They might do it in an emergency but not as a regular dough. In your case, if you want to stick with your present dough recipe, you could try using 1/2 teaspoon ADY. The rest of the ingredients can remain the same. The ADY should be rehydrated in a small amount of the formula water (about 1/4 cup) at about 105 degrees F for about 10 minutes. It can then be added to the rest of the formula water, which can be cool, especially if it is in the 90s where you are. After you have divided the bulk dough into the desired number of pieces and shaped them into round balls, you should brush them with a bit of oil (vegetable oil or olive oil), put them in separate containers, and cover the containers with their covers or plastic wrap secured with rubber bands to the containers. You should look for roughly a doubling of the dough balls before using. I can't tell you exactly when that will happen in your particular environment, but I estimate that it should take several hours. You might find that you will have to do some experimenting to find the best amount of yeast to use. Even then, it will change as your kitchen temperature changes over the course of the year, along with the outdoor temperatures.
If a cold fermented dough appeals to you, you might want to read the following thread, for a NY style pizza dough: http://www.pizzamaking.com/forum/index.php/topic,2223.0.html
. I don't know if that style will appeal to you and your family, but it is a simple and reliable dough and there are many helpful tips on pizza making in that thread, including tips on hand kneading, at Reply 65 at http://www.pizzamaking.com/forum/index.php/topic,2223.msg63786.html#msg63786
. In reading the abovereferenced thread, you might also identify things that you have been doing wrong.