Your pizza looks a lot like mine when I first got into making pizza. Doing a 3 day ferment is definitely the right thing to do, I think 5 days is ideal. Have friends or family who throw away the end crust when eating pizza? Fermenting the dough 5 days will change this. I've done 1 and 2 day ferments and that is really to short of a time to make any difference. When you ferment for 5 days, there's definitely a difference and you can smell it when you open the container. That smell will still be there when you bake the dough.
I've been experimenting a lot with pizza dough over the last 3 months (rather than just doing 1 standard recipe). In my opinion (I'm still new to this so take it with a grain of salt) it is very difficult to brown high hydration dough with short bake times less than 7 - 8 minutes, unless you have a brick oven. Stone can cook the bottom crust quick enough, but the ambient temp needs to be a lot higher IMO to cook 67% hydration dough. The only way I've been able to do it is by combining my baking steel with a gas broiler with the pizza on the top rack. Problem with this is that your toppings can burn.
A lot of NY style pizzas have hydrations as high as yours, but they're cooked in ovens that provide a lot of top (and side) heat rather than a ton of bottom heat from a stone in a home oven.
Are you aiming for a pizza that is flexible, yet still has a nice brown crust? I think your stone temp is actually probably too high to let the end crust brown before the bottom burns, and there is too much water to let the crust brown in 5 minutes, even with all the added sugar. I would suggest...
-reducing to 1% sugar
-doing 3 - 5 day cold ferment in your fridge
-reducing to .5% yeast
-reducing hydration to 64%
-freeze your cheese before topping the pizza so it won't burn under the broiler
-heating the stone to whatever temp you want and then waiting until your broiler is cycled on and then only using the broiler with the pizza on the highest or second to highest rack in your oven.
You could probably cut the oil in half, but I've had successful pizzas with oil as high as 5% and as low as 1% or none at all. When I make recipes I usually take one of my previous recipes and repeat the recipe with one difference and "all else equal". The only real difference I can notice with and without oil is that the dough will not be as tough to chew and be a little more tender. From what I've done, it seems pointless to use over 3% oil in normal ny style pizza. Olive oil is 'spensive stuff and not worth adding that much oil. The only time I would use a lot of oil is when you're making thick crust pizzas. Think about it...if you have a thick crust, that means a lot of chewing, so you want your dough more tender and easier to chew since you'll be eating a lot of it. But if you're making .07 thin pizza, a lot of oil won't really help anything. If you want flavor from it, I used to brush a layer of garlic olive oil on the crust before baking.