Well, that explains why it didn't toss very well.
I didn't get it quite as thin as I wanted, but I would say that the quality was better (even though thick) than my other attempts. I only used half of the dough, so in the next day or two I'll try again (I can only eat so much pizza).
I have noticed that using one particular pre-made mix, I was able to flip it back and forth and let gravity shape it out, but have not been able to repeat that with other recipes. I thought maybe I was just lucky that day, but it helps to know that not all recipes are tossable.
It's back to the rolling pin, I guess. Is there such a thing as a "home sheeter"? I might be willing to spend one or two hundred bucks at most, but I'm not interested in springing for a commercial sheeter.