Sicily, Italy's surrender, and the end of the Suez threat

Did the fall of Sicily by itself end the Axis threat to Suez and make a mainland invasion unnecessary?

The danger ended in this period, but the decisive mechanism was the Allied entry into Italy and Italy's surrender, reinforced by the Italian fleet's fuel limitations.

The prompt argues that Sicily's fall ended any realistic danger to Suez and therefore weakened the case for invading mainland Italy. The threat did disappear at this stage, but the sequence matters. Allied troops entered mainland Italy, the Italian government surrendered, and the Italian fleet ceased to be an active strategic threat. The fleet had considered fighting, but fuel shortages constrained what it could do even before the surrender. The exchange separates the strategic result from the political and military actions that secured it.