William the Conqueror didn't inspire Norman adventurers in the Mediterranean. They were present in the region long before William was born.

Normans had been fighting as mercenaries in the Mediterranean region both for Lombard counts and the Byzantine empire since the 900s. That was the origin as well for the branch of the Hauteville family that eventually ruled as Kings of Sicily. They had come to Italy as mercenaries / adventurers around 1035, only a few years after William the Conqueror's birth. Robert Guiscard became Duke of Apulia in 1059, six years prior to the Norman conquest of England.

The Normans had been busy carving out fiefdoms in southern Italy at the expense of the Lombards or Byzantines since William had been a child. The conquest of Sicily occured after William, but a Norman military presence had been a fact of life in southern Italy for many years prior, and they had already been carving out fiefdoms that were putting them on a trajectory for conflict over Sicily.

Side note, Robert Guiscard is the most interesting Norman warlord IMO, despite William having greater fame, and his wife Sikelgaita (although she was a Lombard, not Norman) was also a badass. She sometimes accompanied Robert on campaign, commanded at the siege of Trani, and brought reinforcements to Robert while was campaigning against the Byzantine empire. At the battle of Dyrrachium, while in full armor, she railled some of Roger's troops as they wavered following a Byzantine repulse. The Italian Normans were also playing in a far more interesting historical sandbox with a more diverse and interesting cast of characters. Byzantines! Lombard counts! The papacy! The emirate of Sicily! Plus Norman Sicily, at least early on, is fascinating in that it was one of the more tolerant medieval states & produced some interesting cultural exchange between the Normans, Greeks, and Muslims of southern Italy & Sicily.