This is pretty much the idea. At the start of the game Imperial authority had been deteriorating for a long time to the extent that successive emperors had failed to exert control over the area. Full HRE membership was never really a good way to handle this region as the Italians didn't participate in Imperial politics like the Germans did and the Emperor was evidently not prepared to defend them the same way either. If you want to turn around a process that has been going on for hundreds of years you need to do something pretty drastic (and HRE emperors in conflict with the Pope is not by any means unreasonable given the history of the Papacy and the Imperial Throne). It's not necessarily going to be worth it but it is possible.
Wiki sums it up ok I think http://en.wikipedia.org/wiki/Kingdom_of_Italy_(Holy_Roman_Empire)#Staufer (if the link doesn't work it's everything atleast from the section called Staufer and onwards).
The only emperor in our timeframe that really seems close to be able to restore the status of the Empire in Italy was arguably Charles V (who incidentally sacked Rome, even if it wasn't the original plan) but rather than entrenching his Italian gains in the Imperial system he chose to let them pass to Spain (you'll recall he ruled both Spain and Austria at the time).
If you want to translate that to game terms (which is always a bit perilous but still) you could say he'd be conquering Italian land after the area had left and then made the choice not to add them back into the Empire.
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