“Italy campaign did tie up German troops...”
That is true. But you never invade that hilly country from the bottom. Even the Germans told the allied commanders that after.
If they would have cut them off up top, they would have isolated the whole dam works.
The Allies were concerned about the reach of their air power. Although they had conquered Sicily, there were not enough airfields available on Sicily to ensure that they could control the air space over the landings. Tunisia was quite a bit further away so aircraft taking off from Africa would have little loiter time over the battlefield. Salerno was chosen as the landing location as they felt they could command the air at that location, but that any landing further north was a much dicier proposition.
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...