Leaving Sicily be allows the axis to put pressure on shipping through the Strait of Sicily. You might be able to do Sardinia and Corsica instead of Italy (since the Axis will likely see Italy as the next target, they probably won't have too many forces there).
From Jacksons 'The Battle for Italy' & Atkinsons 'Day of Battle', Pogues biography of Marshall, and sundry bios of Eisenhower...
The decision of Sardinia (Op Brimstone) vs Sicily (Op Husky) was settled at the Symbol Conference at Casablanca in January 1943. The assistants to the Allied Combined Chiefs of Staff were directed to lay out a broad set of options in the Mediterranean for 1943. Their recommendations were for Sardinia/Corsica first in March or April, then Sicily in the summer, and other operations in the Balkans imeadiatly following. Operations against Italy directly were not specified.
Brooke objected to the priorities and argued for Sicily first, leaving other operations in the Balkans/Italy/and Sardinia aside for the moment. Churchill accepted Brookes arguments. The US leaders dont seem to have cared much either way. At this point the assumption was the Sicilian attack would occur in the spring in the March-May time frame, following the assumption the Axis would be expelled from Tunisia. That was before Eisenhower arrived and reported the Tunisian campaign would not be ended until April or May.
Despite this probable change in schedule Sicily was retained as the next target and Op. Brimstone vs Sardinia left of the schedule. Despite effective Axis air attacks from Sardinia on Allied ships in the eastern Mediterranean and its potential threat to northern Italy Sardinia/Corsica were off the table.
Eisenhower had little to say about this as his command was confined to Morroco/Algeria/Tunisia and broad Mediterranean was outside his assignment in January 1943. Churchill, Roosevelt, and the Combined Chiefs of Staff reserved that for them selves at the January conference. Ike was on his way back to Algeria before the Symbol conference was completed. Shortly after Ike had his responsibilities expanded to include the east and central Mediterranean.
Responsibility for Planning Op Husky against Sicily was handed over to Ikes HQ at the end of the Symbol conference. As was responsibility for the other vaguely identified subsequent operations. The air commanders in the Med. begin lobbying for capture of Sardinia/Corsican airfields for their use, as well as for airfields in Crete, Italy ect... That was something of a distraction which was increased by others lobbying for further operations in the Balkans. Ike directed a planning cell for Op Husky be established in his HQ, and pondered the selection of commanders, Patton became the first candidate for a US component & Montgomery was eventually chosen over Anderson, who was rapidly falling out of favor with Alexander, Ike, and others.
Planning for Op Husky was split up between Ikes HQ, Alexander as Ikes ground commander, the 8th Army, and the newly formed US 7th Army.