Here's a good video on forming.
Pay close attention to the stretch he does at 1:32. Also, as you're pressing out the crust, I find it helps to keep the middle from thinning, if I start with a little mound there.
We get quite a few members who wistfully daydream about eventually owning a wood fired oven (WFO), but are not aware that typical home ovens can, with the right setup, produce close to WFO results. In your particular instance, where all your favorite pizzerias use relatively longer bake times, your need for a WFO is even less.
Now WFOs have an incredible beauty and mystique, so I definitely am fully behind your quest to own one, but, as far making an Aniello's-ish pizza (and better), the right stone will do that for you- and not a year or two from now, but now.
Tell me a bit about your oven. Gas or electric? The dial goes to 550, correct?