Hand grating cheese yourself is definitely better IMO. The preshredded cheese you buy at RD or the supermarket contains anti-caking agents that are not good for you and sometimes imparts an off texture to the cheese.
It is possible that the pizzeria you go to blends aged mozzarella with a small amount of provolone or white cheddar, but I'm not certain. When I was able to get into Restaurant Depot with a temporary pass, I bought their house brand Supremo Italiano three cheese blend, which is a blend of 80% mozzarella, 10% white cheddar and 10% provolone. It gave my pies a richer flavor profile. But then I learned about the additives in the preshredded cheese and stopped buying it. You could certainly make your own blend
There is a pizzeria around the corner from me that blends whole milk and part skim mozzarella on their pies.
I hope this helps you.