I don't know why I didn't see this until now, but I thought that first part sounded familiar. Specifically, it felt like you were doing what I've done with my Tommy's clones. Now I'm curious to see what happens in the subsequent replies, which I haven't read yet.
Yeah! I tried doing the laminated thing for a while. I could tell that it made a difference. It was a much more tender, softer crumb. Still crisp, but very light. Not exactly flaky like a cracker crust (maybe I did something wrong), but I can always feel the difference when I do the lamination like you talked about.
I stopped doing it, though, because I ended up not being as into the texture. Like I said, I may have done something wrong, but I wasn't getting Chicago/Midwest cracker-crust type flakiness like in your pics, just a texture more consistent with a shortened dough. In the end, I decided going down that path might delay me, so I went back to doing the recipe the way I'd started.
ALSO ... I'm not doing a 55% hydration, which doesn't have as much snap-back as the lower hydration I was doing before. And once I went back to using Crisco to line the pan, that helped too. I was using olive oil, which is too much of a lubricant and allows the dough to contract too much. The Crisco allows the dough to grip a little better...
Anyway, I'm following a formulation similar to Norma's in this thread and doing more of a Star Tavern–style pizza again, with some tweaks.