Tranman, cold fermentation develops a considerable amount of gluten, so it's very difficult to underknead a cold fermented dough.
You seem very earnest in your quest and I don't want to dampen your excitement, but I think, in your effort to improve your crust, you're going in a few too many directions and potentially overcomplicating things.
Although I'm a fan of punch downs and stretch and folds, I think they're a bit experimental for NY style pizza, and if you don't understand them well, they can be especially counterproductive. Gluten can only expand so far and then it will break. If you're working with an extensible flour, you get a little more leeway in how much you can work your dough, but you still want to be careful. Cold dough is especially fragile/prone to tearing. Also, if you work dough too close to forming the skin, it can make it hard to form and give you a tough crust.
I'm also a big fan of high hydration, but 68%+ is pretty advanced pizzamaking. Doughs that are that slack can be very difficult to work with.
I also think that although starter could make for superior crusts/breads, again, I think it's something you want to get into once you master the basics.
I'm not saying don't go above 68%, don't do punch downs and don't use starter ever, I'm just recommending to avoid these more advanced practices for the moment and stick to the basics for NY style pizza making.
Speaking of sticking to the basics, long cold ferments are usually not something your local pizzeria might do, but those are very much proven to enhance crusts, and, as long as you keep your kneading in check and don't go too crazy with your yeast, 3 day cold ferments are an easy way to bring more flavor to your crust.