I think you have zeroed in on the potential sources of the problem.
The TJ dough formulation you have been using looks to be a good one. You will note, for example, that the sequencing of ingredients, by their predominance (by weight) in the formulation, is the same as the one I posted, except that the TJ formulation apparently uses a bit more olive oil than the clone formulation I posted. But those differences are likely to be inconsequential. If you decide to try the clone formulation I posted, and if you decide to use bread flour, the King Arthur brand should be a good choice. The KA bread flour has the same profile (list and ordering of ingredients, including the barley flour) as the flour used in the PJ dough.
I think the age of the dough may have been a factor in the results you achieved, but now that I know what kind of pans you have been using, I think you definitely want to get away from them. I have seen ads for insulated air pans for baking pizzas, even ones with a dull finish, but my recollection is that they are primarily for reheating frozen pizzas. I would avoid them. Given a choice, the only pans I would use for pizzas, other than old, dark, well seasoned ones, are the dark anodized ones, including the cutter pans and the perforated pans/disks, as sold by pizzatools.com. At least with the solid pans, you can oil them and get very good bottom crust browning.
If you decide to go with a pizza screen, I would put it on the lowest oven rack position, but be prepared to move it up toward the middle or top if the bottom crust browns too quickly. If you like a crispier crust, you can proof the dough before dressing (to let it rise) and use a lower oven temperature (e.g., around 435-450 degrees F) and let the pizza bake a bit longer than normal. Each oven is different, so you may have to do some experimenting.
Maybe sometime you can try the TJ clone formulation and compare the results with the dough you have been buying from TJ.