There is an interesting documentary on Netflix about Churchill's advocating for attacking what he called "Hitler's Soft Underbelly". I believe that it can be said that Marshall advocated for a channel crossing as soon as possible but Churchill argued against it due the specter of Dunkirk and British losses in WW l.
The Brits position was no doubt much enhanced by Monty's success at El Alamein and the American's receiving a bloody nose in their first time out in Africa.
It was said that the soft underbelly had become the Crocodile's tough snout with the allied armies mired down slogging their way up the boot. Monte Cassino proved to be a classic example of facing an entrenched enemy who possessed command of the high ground.
Regardless of the correctness of the strategy of invading Italy one could say that the American's were well battle hardened for the final assault on the continent.
Interesting.
The only division I believe that saw action in Sicily and then went to England for the Normandy invasion was the 1st Infantry Division (1ID) and the 82nd Airborne Division. Most of the troops used in Normandy had been sent directly to England and then straight to Normandy.
There were three divisions that were withdrawn from action in Italy and used to invade southern France (3ID, 36ID and 45ID), but most of the troops that went to Italy got battle hardened in Italy and stayed in Italy.
Churchill was a great war PM, but this "soft underbelly" stuff was garbage. A huge waste of time, treasure, and blood. (Which is NOT to say the men who fought there were not brave and doing their best, just that their efforts did not contribute very much to Germany's defeat).