Has Sicily always been viewed as one with the rest of the peninsula? It seems like throughout ancient and early medieval history, Sicily was consistently somewhat independent of the powers of the Peninsula. Rome had to conquer the Greek city states in the later 200s BCE. Sicily was controlled by Islamic powers in the early medieval period, then the normans, etc.
The fact that sicily enjoyed different invasions doesn't mean it wasn't considered an Italian extension by most politicians and historians throughout the Ages.
The greeks had colonies throughout the south of the Italian Peninsula, sicily was just their last bastion. Romans worked their way through southern Italy and conquering sicily was just the next obvious step in a century long struggle to control the region.
When the west Roman empire fell, the eastern Roman Empire (see: Byzantine Empire) took control of southern Italy and Sicily. After three centuries of struggles against germanic tribes and other powers, the Arabs came and conquered Sicily.
Every kingdom in Italy tried to take it back, but the deed was accomplished by the Normans, another foreign invasor.
Then Sicily was inherited by the Holy Roman Emperor, Frederick II, that kept his court often in Palermo and incentivised a flourishing in culture and literature: the Sicilian School of literature is considered as one of the most important in Italy and a source of inspiratuon for many artists throughout the centuries. Then other powers came and went, but all of this to say that Sicily was always considered a part of Italy, an extension of the Italian Peninsula.
After ww2 there was a short lived "indipendentist front" but it was a mixture of hate for the central government, pro US sentiments, anarchic uprising, mafia and political machinations.
If you want to look at an Island who isn't all that Much Italian, look at Sardinia and Corsica (that aren't that much Italian but surely aren't french either)