Well, I think your question sort of answers itself. Italy wasn’t unified after the fall of the (Western) Roman Empire because there were multiple claimants to the territory.
In the Gothic War alone, you had the Byzantines (backed up by Huns, Lombards, Slavs, and other foreign allies) against the Ostrogoths (who were in turn supported by the Franks, Burgundians, Alamanni). However, that statement makes things seem way more straightforward than they actually were, because those allies often became their own sides, as we see from the Franks, who turned on the Ostrogoths and almost succeeded in defeating both them and the Byzantines if it hadn’t been for a very timely outbreak of dysentery, and from the Lombards, who started out as Byzantine allies and ended up conquering most of Italy soon after Justinian’s death.
And complicating the “historical birthright” narrative is the fact that the Ostrogoths were legally allies of the Empire, a lot of their leaders had received a Roman education and worked hand-in-hand with the Roman Senate, the surviving Roman bureaucracy, and the Catholic Church (despite the fact that they were personally Arians rather than Chalcedonians), so could arguably be considered to have as much of a claim to said birthright as the Greek-speaking, Orthodox Byzantine invaders.
