Was that ever true of Italy as a whole? Or was it like the Yakuza, which exists but doesn’t affect your average person, at least in recent memory.

the Yakuza isn't running local or national governments

in Italy, there are deep ties to governments, esp. in the south, and they've had their fingers all the way into national leadership before; see also: bunga bunga

https://en.wikipedia.org/wiki/Propaganda_Due