Sardinia/Corsica 'Operation Brimstone' were to be taken in October/November 1943. Brimstone was originally proposed at the January 1943 Symbol conference as the next Allied move, but Operation Husky was chosen instead.
Once the decision was made to invade North Africa, General George C. Marshall knew that an invasion of France in 1943 was off the table. So after Torch, we did Husky, the invasion of Sicily. After Sicily, the next logical step was to invade Italy, to knock Italy out of the war, to gain air bases and naval bases, and to try to stretch German military assets.
Marshall did make another fight for France in 1943 during the Symbol Conference (Casablanca) in January 1943. He and General Alan Brooke Chief of the Imperial General Staff went at it for a couple days over the issue until Roosevelt and Churchill decided for Italy. After that the discussions were over what approach to Italy. Sardinia and Corsica were actually proposed by the British to be run first sometime in the late spring or summer. Though after much more talk Husky was chosen instead as the leading operation, with Sardinia pencilled in for later.
One of the sources of friction during the Symbol conference was Eisenhower had to reiterate bad news about Tunisia. Although everyone had seen his reports many came to Casablanca hoping there would be good news awaiting from Ike. Instead he had to tell them it would be two more months before enough airpower was established far enough forward to wrest air superiority from the Axis. That and the poor transportation between the Algerian ports and Tunisia meant Ike could not guarantee driving the Axis out of Tunis/Bizerte before May. That complicated the discussions over strategy for 1943 as the number of variables emerging over six months reduced it to the level of a history forum What If discussion. At the end of the Symbol conference Ike was handed a schedule of midyear operations based on guesswork and told to 'take care of it'
Brimstone was set aside again for the execution of Operations Baytown and Avalanche. The USAAF wanted Corsica as a base for its medium bomber, and by January six wings of mediums (500+) were based there, plus fighter wings. The French government wanted Corsica as a source for recruits, and a base for covert ops in southern France.
Operation Anvil was originally pencilled in for March or April 1944. The intent was to use it to draw German reserves away from northern France in preparation for Operation Neptune. Op. Anvil was postponed to boost the number of landing craft available to Operation Neptune. The need to sustain the Anzio beachhead also drew down the amphibious transport available.
Leaving the Italians to work out their own problems could cut three amphibious assaults out of the Allied schedule in 1943. Avalanche, Baytown, and Shingle. This would allow Operation Brimstone to be executed earlier in September, or even late August. With no attack at Anzio there is a possibility for executing Anvil earlier allowing the amphibious fleet to be transferred to the UK in time for Neptune.
There is another factor to be considered. Marshall Pietro Badoglio and his government would never agree an armistice with Allies in September 1943 as long as Allies armies wouldn't land on Italian peninsula itself.
Simply put, the surrender of Italy was conditional upon a landing on the mainland. That was what had been negotiated. An argument could be made about the Italian will to fight (or lack thereof) for the Axis from 1943 onwards, but nevertheless, without a landing on the mainland, Italy stays in the war. While the Italian war industry wasn't spectacular, they were putting out some good quality equipment by 1943.
To look at the Italian campaign in a slightly larger picture, to include the Balkans and southern France. Most importantly the numbers of Italians engaged in occupation/anti-partisan duties in those areas, and the need for Germany to replace those numbers once Italy had. The campaign in Italy didn't just draw German troops into Italy. It also drew them into the Balkans and southern France, and northern Italy was well for anti-guerilla duties.
The Italian campaign wasn't a forgone conclusion. Unfortunately, it seems the Allies didn't anticipate the quick German reaction to the Italian collapse, and the subsequent occupation. Had the Allies launched a landing 1 or 2 weeks sooner, we might be talking about a completely different campaign. A quicker Allied reaction likely would have required end-run landings near Naples and Anzio, with the Anzio landings dashing for Rome in connection with an airborne drop. The Italians would have put up little or no resistance, with the landings part of the negotiated surrender. If done 'right', the Italian campaign should have happened differently than it did.