Armed Forces, everywhere in the world, have a military "Chief of staff" but not a military "Commander". This is in contrast to armies, army corps, divisions, squadrons, fleets, etc. which have a Commander and, subordinated to him, a Chief of staff, who applies the directives of the former and turn them into operational orders. This same dichotomy exists in the US Government, with the President and the Chief of Staff, or the French Government, with the President and the Prime Minister, etc.
The Commander of the Armed Forces is defined by the constitution of each country, but usually he is the Head of State. In constitutional monarchies, then, not rarely this power his delegated to a member of the executive power, be him the Prime minister/President of the council or the Minister of War. Until the end of WW1 in Italy the delegated commander was the Minister of War, but this choice was deleterious during this conflict, because this minister hadn't the political strenghth nor the authority to exert his will on the Chief of Staff of the Army, i.e. Gen. Cadorna (his successor, Gen. Diaz, was more malleable towards the Governement, but not much), who led the war almost as he pleased causing huge losses that didn't bother him, as a militar and not a politician, who instead should see soldiers at least as voters if not as citizens. Add to this the fact that usually in Italy the Minister of War was a general himself, but subordinated, within the hierarchy of the Army, to the Chief of Staff, both as rank and seniority. The choice to delegate the President of the Council, i.e. Mussolini, to the supreme command was, therefore, perfectly normal and understandable.
The problem laid in the fact that, when Marshal Badoglio was Chief of Staff of the Supreme Command, he hadn't the mere availability of officers and men to prepare any serious plan nor the will to exert any influence on Mussolini's choices. Choices which, after nearly 20 years of successes, were hardly put into question and, when Badoglio did it (because he did it: he wasn't spineless, just lazy and too old), political rationale was always put in the first place (i.e.: information about a coming German armistice with UK, or a planned coup in Athens, etc.), just like had happened during the War of Ethiopia, the Spanish Civil War, and the occupation of Albania. Gen. Cavallero fully changed the situation, and Mussolini's retreat from any tactical or too technical decision after the setback of the Greek Campaign, allowed the new Chief of Staff much more freedom of action. Then, of course, when high political decisions were taken Cavallero had only to accept them and turn them into practice. Decisions which were never made by Mussolini alone, but always after having informed the King and probably having talked about them with him (they had a meeting every Monday for more than 20 years). The Chief of Staff, moreover, had direct meetings with the King, but everybody involved in these meetings kept their lips closed, due to their respect towards the Sovereign. Even during the Italian Social Republic, Mussolini did not disclose anything about his relation with the King and the latter, in his lost memoirs, apparently wrote some positive remarks about the Duce (according to the testimony of the only man who had read them, sen. Bergamini).