Pizza crust too hard in home oven, how to make it softer

I recommend starting with a chilled dough, puncturing with a knife or fork (see below), and cooking in a greased pan initially then bricking.

Cooking directly on the stone the entire time works better for thinner crusts than mid-dough. By mid-dough, I am referring to the ones that plump up a bit and finish between 1/4" - 1/2" in thickness. They can have both uniform, almost cake-y cross-sections, or air pockets and bubbles depending on how kneaded they are (more kneading means cake-ier, denser dough).

I worked in a shop that made a mid-dough similar to what you describe. It was a standard dough recipe (flour, water, yeast), but I have had success replicating it at home at 450'F with beer and less kneading. At home I use a pan and then finish either on the rack or a stone. In any case, you get a dough that tears nicely and has a good chew.

The dough itself was mixed in the morning and kept refrigerated for as long as 12 hours as individual shells in greased pizza pans covered with saran wrap. To prepare a pizza, we pulled the shell, punched it with a fork (puncture a ring around the edge to create a crust, puncture the inside to allow for air). Dress with sauce and toppings.

We had a stone-bottomed oven, but initially the pizza is baked in the oven in the pan until the bottom of the dough has hardened enough to get "bricked" (the dough should be rigid enough as to be removed from the pan with one spatula). At this point you remove the dough from the pan with a spatula and place it directly on the brick. Cook until the top of the dough browns, and the cheese and other toppings are evenly browned.

For a deep-dish pizza, around 425°F is right, and so is 20–30 minutes. That's starting with cold dough (need to keep the butter layers chilled, at least for a Chicago-style pizza).

Cooking in an aluminum 3" deep cake pan is fine. I suppose cast iron should work too (though it'll heat slower, so might take longer). As has been pointed out in comments, the cast iron much greater heat capacity may be part of the problem; I'd guess preheating it would help. (You can just put the cast iron skillet in the oven as the oven heats, though depending on how long you let the oven heat, you may want it in for only part of the time—no idea what the optimal temperature for it is). Make sure to have plenty of oil under the dough, and also cook on a lower oven rack.

I'd guess that you're using too-watery toppings. The tomato sauce should be pretty thick, much thicker than you'd ever use on a thin-crust pizza. Vegetables may need sweating to get some moisture out. If nothing else works, partially cook the crust (say, ten minutes or so) and then add the sauce and toppings.

I can vouch by Cook's Illustrated's Chicago-Style Deep Dish Pizza recipe. Normally they have a paywall, but currently that recipe isn't behind it, so grab it quick.