> what part of the rest of the pizza-making process can you do better than the restaurant?

A determined home pizzaiolo can make a perfectly decent pizza. But you are right about the temperature. Household ovens simply can't reach the temperatures used in a proper pizza oven, about 700F.

I have had success with heating a large cast iron pan on the stove top to ~700F, dropping the dough in that, adding toppings when it's ready and then finishing it in a broiler or well preheated oven. These guys show how it's done, without words, better than I can describe: https://www.youtube.com/watch?v=HXAW2GseICs It is pretty damn good technique. They're using 00 flour, high protein (available at Italian markets in any big city).

One thing you can do at home better than the pizza shop, however, is to use crazy good artisinal toppings, dialed in exactly how you like it.