Unless I use it right away, my cheese tends to be too soft to work in the KA or Cuisinart, even if chilled thoroughly in the freezer. I have one brand of brick mozz, Biazzo, that's extremely firm, and does work for mechanical grating, but... it has a tendency to curdle as it bakes, so I have to avoid it.
As far as the thread goes, good NY style pizza is all about the butterfat that the whole milk cheese gives off as it bubbles. At least that's my impression of good NY style pizza
Brick mozz just isn't going to release it's beautiful buttery-ness unless it's shredded. Hand formed mozz- that you can cube or slice because it melts a bit easier.