Ken, you are right in that Axis pundits claim they tied down Allied forces and Allied pundits claim they tied down Axis forces. The problem is, do we just count the divisions on the firing line, or do we add in the divisions garrisoning rear areas that the Germans would have had to have there anyway? If those rear-area forces are taken out of the OB, the Germans held up the Allies with about 1/2 their strength. Then, at least some of the Germans fighting in Southern Italy would have had to be there anyway, as garrisons.
After the Fall of Rome the Allies withdrew French and US units forDragoon, so it did become a bit easier for the Germans to delay what was left. Also, the Brits were pulling troops out for Winston's schemes in Greece and the Aegean.
So it's pretty complex. We can't even be certain that Italy would have capitulated in 1943 without Sicily and Salerno, although I think there was a pretty good chance. Italy was definitely a drain on both sides, although I think the Allies got drained more. Normandy was done with troops who were bright green (and not just from seasickness), because the combat-experienced troops were in Italy.
You are right in the fact that both sides devoted units to the war in Italy that could have been used at better places. But I still believe that Germany was suffering more from it, then the Allies, as the Allies had enough troops ready, while Germany was getting short on heavy forces. Just imagine what 5 divsisions of gemran infantry and armor could have done at Kursk. Or how 5 additional and good trained and equipped divisions could have made Overlord much more bloody for the allies. Furthermore the war in Italy forced the Axis to garrsion Austria and Northern Italy much earlier in the war.
And the there is the question if Mussolini would have fallen without Sicily and if the Germans would not have been able to get the Italians back under full control with out an allied invasion and install another facsit leader. Which would have possibly meant that they would have been able to even free up more divisons, as italian fascists could have controlled Italy and put down any uprising.
I wonder why the war in Italy is so often discussed, while Dragoon is not so often focussed on.