Except this would be like if you referred to the mafia as "crooked Italian."
Yes. But I think that's Giuliani's point: for those objecting to the use of the term "mafia", it is effectively equivalent to "crooked Italian", and if you are a decent, law-abiding Italian, drawing a distinction between those like you and those like them is not a bad thing. On the contrary: it's an important distinction to make, as otherwise you run the risk of all Italians being painted as being crooked, which clearly is not the case, just as not all Muslims are murdering terrorists. Why we spend our precious resources trying to make sure that terms like "jihad" and "sharia" are kept out of public view and avoiding the use of terms like "radical Islam" is beyond me.
Except it wasn't. There were plenty of major non-Italians in the mafia/organized crime.