WWI and its outcome undoubtedly played a key role in the rise of fascism in Italy. While the factors that brought to the infamous March on Rome (1922) are multiple and diverse, in my answer I will focus mainly on two points: the attitude of army veterans in the aftermath of WWI and the evolution of Italian national discourse before the rise of fascism.

Although Italy had won the war, the outcome of the treaty was, in the eyes of many, unfavorable for the country. Some of the lands Italy had hoped to claim by joining the war (the so called terre irredente) were indeed granted, such as the city of Trieste and the Trentino region, but the Entente powers refused to acknowledge the Italian claim on Dalmatia, in the Balkans. This treatment was deemed humiliating by those who had pushed for the war in the first place (the most famous of whom, perhaps, was Gabriele d'Annunzio, a novelist who would then lead his own militia to occupy for a short time one of the contested cities, but that's another story). Many of those who had fought in the war, especially the officers, also felt that the country had been treated unfairly at the peace conferences and that, like you said, the territorial gains were disproportionate to the tremendous sacrifice the war had imposed. But the veterans' resentment was more complex than that. Many Italian officers, even the Chief of Staff Luigi Cadorna himself, had developed during WWI a lack of trust towards the politicians and the state in general. They often complained that the socialists and the giolittiani (those who supported Giolitti, a politician against the war), and all of those that in general did not approve of the conflict were actively sabotaging the war effort through harmful propaganda which, in their eyes, produced demotivated and unreliable soldiers. These suspicions became even stronger after the disastrous defeat of Caporetto in 1917. The idea that this military disaster was directly caused by inept politicians took hold. Moreover, when veterans returned home, they found a country plagued by social unrest, which brings me to my second point.

As historian Mario Banti argues in his excellent work Sublime madre nostra, the rhetoric used by those who had advocated for war in 1914 was very similar to the one that had characterized the national discourse of Risorgimento, the Italian independence period: such rhetoric was centered around the ideas of sacrifice, duty and the almost "sacred" nature of the nation. This set of ideas was being seriously challenged by socialist ideals in the immediate aftermath of WWI and many veterans felt that these new ideas, the uprisings, the strikes and the social unrest almost "betrayed" the ideals for which they had fought and suffered. They felt underappreciated for their efforts in the war. Early fascism "revived" the original national discourse, and Mussolini was seen by all those who were in some way displeased by the outcome of the war as the man who could set things right. To quote a young veteran who in 1919 writes about Mussolini and fascism: "This movement [fascism] shall be [...] the beginning of a new age, a better one than this." I think you are right when you say that Italy being a young nation somewhat favored the rise of fascism, as Mussolini was able to employ an already established set of values, the ones of Risorgimento, in order to twist them to his own advantage and to show himself as the defender of order and traditional values. The veterans' anger and disillusionment and the lack of trust between military and state obviously played in his favor.

There is much more to say on the topic and many more reasons that allowed Mussolini to take control of the country, but I hope this answer showed the strong connection between the outcome of WWI and the rise of fascism.

I refer you to the thread from last month titled What was the difference between fascism's rise in Germany and fascism's rise in Italy? featuring the work of /u/commiespaceinvader and /u/Klesk_vs_Xaero and /u/restricteddata