To get back to the original question. Invading Italy was in my eyes right for political reasons. What isn't mentioned yet is the constant pressure from Stalin to open the several times promised second front. As
@McPherson already mentioned there were no other good options at that moment. Preferably in Stalin's eyes was France, but that would have been out of the question for the Wallies at that time. The Balkans (Churchill liked that option) would have made Stalin extremely suspicious. And what would really be gained by Norway. The Italian campaign didn't go as planned, but even that failure offered several advantages to the Wallies as already discussed by others.
The OP's question is open-ended, and it forces us to clarify the premises.
Do we assume that the rest of the ETO's history has played out exactly as it has OTL, right up to HUSKY?
If so, we're too far along to try ROUNDUP - so we are, indeed, looking for some kind of op to keep the Germans on the backfoot and Stalin from erupting.
Of course, at this point, Badoglio has entered negotiations for a surrender. If the Allies handle that more aggressively, a lot more real estate could be gotten for free than was the case OTL. But really, this is just getting me to what I think is the unasked question so far: Was the invasion of Italy
we actually got historically necessary?
Consider the alternatives:
1) The Allies could immediately follow up HUSKY with operations for securing Sardinia and Corsica, paired with no more than a minimal move into Southern Italy via BAYTOWN and SLAPSTICK. This puts virtually all of Italy within both fighter and bomber range, and forces the Germans to deploy forces up and down the peninsula against possible landings - or, they just retreat to Northern Italy on the Gothic Line, or even further (though this was against Kesselring's belief).
2) The Allies could immediately follow up HUSKY with AVALANCHE, BAYTOWN, and SLAPSTICK, but leave nothing more than a credible holding force at the Volturno Line, with no further offensives planned for the theater. No meatgrinders on the Moro River, or Cassino, or Anzio. This still gets them their air base at Foggia, and still ties down Tenth Army more or less as happened OTL.
The advantage of either of these alternatives is a great reduction in casualties as well as resources in Italy by declining major combat through horrifically brutal terrain, leaving more available for OVERLORD and DRAGOON the following spring. The Allies suffered over 300,000 casualties in the Italian Campaign (exclusive of HUSKY) and almost all of those were post-AVALANCHE. And let us not forget, 150,000+ Italian civilians killed, too. Can we really say they were worth it? German casualties were not much greater, so we can hardly even justify it just on attrition grounds.
No offense to the men who fought and served honorably in Italy (which includes at least one close relation of mine), but when I look at the Italian Campaign as it actually unfolded, what I mostly see is a terrible waste, a campaign waged at high cost over horrible ground, to little obvious strategic benefit.