I think southern and central regions of Italy were not suitable for mechanized warfare...
Northern Italy was little different...In 1945, Allied troops advanced very fast in April offensive...
If the invasion would have taken place at the north of Italy and started cut cross the area, the Germans would have had to evacuate or become trapped. The only way they could have got out is across the water to the east. Then they would have had to face the Russians eventually.
Of course Hitler, in his genius for strategic war, would have insisted upon holding the ground in Italy. The result would have been complete entrapment and taken Kesselring’s entire army group out of the war.
The Germans would have had to contend with the Italians as well as getting attacked from the air every time they showed their faces.