Destroy Third Reich from which direction was a great debate among allies. For Soviet Union, this was not a question since she always advanced from the East. However, for western allies, they had at least 3 choices.
The first choice was landing on North shore of France, this is the most straight forward way, but it is hard. The second choice was from Italy, but this way was too far and Mountain Alps was a tremendous barrier for advancement, this was proved to be true at the end of war. The third choice was from Southern France, although this way was easy, it was too far away from main battle field, western allies army might be lag behind by Red Army and Soviet Union might occupy the whole Central Europe at the end of war.
What was the choice of Western Allies? In fact they did not choose any place but landed on all the 3 site. Ally landed on Salerno, Italy on 3 Sept. 1943 in Operation Avalanche, then they landed on Normandy on 6 June 1944 in Operation Overload, the final landing was Operation Dragoon on 15 Aug. 1944 at Southern France near Marseille. So Western Ally DID try to invade Germany from the South through Italy, and as you can see, invasion of Italy started the earliest.
However, Battle of Italy was very difficult and unsatisfactory. Most of British best troops and a large US army were trapped in this battlefield, the advancement was very slow. Although Ally captured Rome on 4 June 1944, it was still half way to Germany, and not counted the army needed to cross Alps. Ally finally stopped on River Po at early May 1945 and still could not invade Germany from the South.
Landing on Southern France finally converged with the Grand Army departed from Normandy and their advancement was much more satisfactory, they cross Rhine on late March 1945 and capture Berlin on early May. Fact proved started from Normandy was the best way to destroy Third Reich.