Was the Italian Social Republic really a German puppet state?

[Kmut00]

There is one thing that has always caught my attention about the current official narrative of World War II but that has rarely been dealt with a revisionist perspective. The issue is about the legitimacy of the Italian Social Republic regime on the one hand after the dismissal of Benito Mussolini on July 24, 1943 and his arrest on July 25, 1943, the capitulation of Italy to the Allied forces on September 3, 1943 (with the creation of the pro-Allied and Antifascist government of Pietro Badoglio) and its proclamation on September 23, 1943 and that of the Badoglio government and the Antifascist Partisans on the other hand.
On the one hand in the Allied propaganda the RSI is always portrayed as a German "puppet regime" without any legitimacy and Mussolini as a mere "vassal" of the Germans from 1943 without any real authority (in fact it is often said that a thereafter the Duce lived in practice under house arrest by the SS, who restricted his communications and travel). On the other hand, the government of Pietro Badoglio and the Antifascist Partisans is described as "legitimate" (despite the fact that it was formed after the Allied occupation of Sicily and southern Italy). As in everything related to the Allied propaganda, I see here a clear display of cynicism and hypocrisy. The Allied invasion of Italy is also described as a "liberation" (and as "proof" of it, they often show the lynching of the corpses of Benito Mussolini, his lover and other Fascist hierarchs in Milan's Piazza Loreto). However, I would like to ask to you the following questions, Is there any truth to all this rhetoric? Did the Italians really see the Allies as "liberators"? Did the most of the Italian people remained loyal to Benito Mussolini after the events of July 1943? How strong was the popular support for the Italian Social Republic? Did the majority of the Italian people consider Badoglio's government legitimate? How strong was the popular support for the Italian Antifascist partisans?