In Dante's Divine Comedy, there's plenty of thinly veiled commentary on how the squabbling of the Italian city-states is pointless and the country should be united, notably when Dante meets Emperor Justinian who bemoans the divided state of Italy at the time. Machiavelli also commentates in his last chapter of The Prince that Italy needs a leader to politically unite the peninsula and throw out the French, Germans, and Spanish who are embroiled in the Italian Wars (While The Prince isn't to be taken plainly at face-value, Machiavelli was a supporter of an Italian republic).
But outside of small circles of educated political theorists and authors, was the idea of a united Italian state popular among the masses of the divided Italian states, or did more regionalist attitudes prevail among the common populace of areas like Naples, Lombardy, Veneto, Tuscany, etc?
This answer will focus on Dante and his age about whom I know much more than Machiavelli, but I can definitely take some stabs at that side of things.
But first, a general admonition: beware of reading thoroughly modern ideas like a united Italian republic into early modern, and in the case of Dante, medieval writers.
While The Prince isn't to be taken plainly at face-value, Machiavelli was a supporter of an Italian republic
Was he though? Or was he in favor of an Italy gathered together in some sort of nouveau Lombard league with a great prince at the head of an army of Italians that could drive out barbarian kings and allow the Italian states to return to some imagined status quo? I would be very careful about asserting that Machiavelli was a supporter of an Italian republic. He was certainly a supporter of Italian republicss, but a united Italy under one prince is a bit harder to prove.
But I digress, I know quite a bit more about Dante and even wrote about the passage you describe in my dissertation. First things first, Dante was not an "early modern" writer. He was thoroughly medieval in his thought and his social conflicts.
So, what was Dante worried about. As Marco Santagata very clearly effectively elucidates in his wonderful biography of the poet, the man was preoccupied with the breakdown in the social order brought about by the introduction of newer families in Florence. While Dante is primarily focused in the Tuscan context, he sees this pattern everywhere as the conversation about Mantua between Virgil and Sordello indicates in Purgatory VI.
What Dante wants is not a united Italy, but for the region to be brought under imperial hegemony, which he believes will bring about the political and social order that Italy lacked at the time. Indeed, Dante wanted Henry VII of Luxembourg to be the emperor who ruled over an Italy that was incorporated into a universal empire.
Why would this train of events bring about stability? The legal thinkers on which Dante and his contemporaries drew based their theories on a new interpretation of Roman jurisprudence. But without an emperor at its head, this socio-legal system could not sustain peace. under and emperor, however, it was hoped that the chaotic clash of orders could be brought to peace. Though not groundbreaking (Dante was a genius poet and linguist but fairly standard in his political thought), the place to look for a more complete explication of this would be Dante's Monarchia, which lays out quite a bit of this thought. But Dante does succinctly state this position in Purgatorio 6.86-93:
"Search, miserable one [Italy], around your shores then look into your heart if any part of you rejoice in peace.
If there is no one in your saddle, what good Was it Justinian repaired your harness? Your shame would be less great had he not done so.
Ah, you who should be firm in your devotion And let Caesar occupy the saddle, If you but heeded what God writes for you."
Here the Ius Comune is the harness that guides the horse that is Italy. But the harness cannot control the horse without a rider, Italy requires an emperor.
Even if you look at the passage from Paradiso 6 that you reference, Justinian only comes to a description of the social ills of Italy after a lengthier description of the Roman empire that will, in turn, explain the failures of both Guelphs and Ghibellines to affect lasting peace anywhere in medieval Italy. For Justinian, the expanse of the Roman Empire over the known world brought about a lasting peace, explicitly referencing the closing of the doors of Janus by Augustus described in the Res Gestae and known to Dante from the Aeneid 1.286-96. The speech follows a succession of world emperors spanning the period from Augustus to Charlemagne and including Justinian. Dante, with Justinian as his mouthpiece, laments the failure of Italy to participate in a new "world Empire" because the Guelfs reject the imperial banner and the Ghibellines seek to claim it for themselves and not for the good of a new imperial order.
This sort of thinking was rather widespread, and while politicians did not perhaps look at the continuum of history with such nuance, we know that Genoa, after around 150 years of incredibly violent factionalism, looked to kings at various points to potentially bring about a lasting concord of orders (they even court Henry of Luxembourg for a bit before rejecting him, a course of events that causes Dante to castigate the Ligurian city) and eventually invite the King of France in to potentially solve the problems of their city.
I'd be happy to answer follow-up questions of clarify any remarks here, apologies if it's hard to follow.