I don’t know about Italian but I can tell you Greek Americans plainly can’t speak Greek.

Same with ‘ethnic’ cuisine, they don’t put lettuce in “Greek” salad because that’s how it was originally but because that’s what was available and accepted in the US.

Its also far more likely that very regional or even just family traditions/customs/recipes got attributed to whole nations rather than an elite managing to wipe it out from the original group.