Here's some thoughts.
The oven spring from a super hot stone is critical to the airy chewy texture of a NY style pizza. That 8-9 minutes- we have to shorten that. How hot does your oven go? How long are you pre-heating the stone?
Besides being a little dense, I think the texture of your pizzas would improve if the crust was a little bit thinner. Are your pizzas close to filling your entire stone? If so, I'd play around with decreasing the ingredients by 10% while still stretching it to the same size. Stretching dough thin and evenly is no easy task. Flour is pretty cheap as is yeast (in the quantities we use it), so you might want to double your recipe and have an extra skin (or two) on hand in case you mess up or just want a little extra practice.
Although there isn't much oil in your crusts, in the pictures, the crust looks a little oily- both top and bottom. Are you forming the skin using oil or with flour?
A big part of oven spring is a dough that's proofed long enough to have plenty of bubbles before forming- and formed in such a way so that many of the bubbles in the rim are still intact. This generally translates into not pinching/pressing the outer rim during forming. It also means that you've used enough yeast so that the dough has approximately doubled in the fridge and, by the time it's warmed up, it's tripled (or even quadrupled). Another important aspect to keeping the bubbles in the dough is using a container, that, after being lightly oiled, will release the dough easily after being warmed up. If you have to mangle the dough to get it out of the container, you lose a lot of the bubbly goodness.
Puffy airy crusts are a result of high heat and elevated hydration. The water creates steam, which, in turn, drives the crust up. After the pie is baked, residual water softens the crust and makes it floppier. KA bread flour used to be incredibly sticky and unmanageable for me, so I stopped using it. I've never worked with KASL, though. Just to clarify, though- dough should be tacky and require plenty of flour to work with comfortably. If you flour a dough ball and leave it on the counter for a couple of minutes, it should stick. If you leave it on the peel for a couple of minutes- same deal. It should be a little difficult to handle/require fairly constant flouring, but not be impossible to handle.