The point isn't that those considered white don't have culture, its that "white" isn't a real culture in and of itself. Being Irish and Italian culture predate the category of white itself. Malcolm X once said there are no negroes in Africa, just Hausa, Fulani, Wolof and so on. Similar with Europe, before the transatlantic slave trade there was no white or black, there was English and French and Prussian and so on.
You can say in response "so is there no such thing as black culture then" but the difference is blacks were cut off from their ancestral culture so they had to construct something new. Whites were not forcefully cut off from their ancestral culture, therefore when they expressed cultural pride it tended to be in reference to that ancestral culture rather than merely whiteness(though they're not mutually exclusive). These days most whites are generally alienated from their ancestral culture but that's more so through slow assimilation and intermixing overtime rather than the more harsh and clean break that blacks were subjected to.
They're not equally cringe though.