The idea of unifying Italy certainly predates Gian Galeazzo Visconti. A generation before, members of Cangrande Della Scala's court wrote that if he had wanted to he could unify Italy in a year. It is fair to think that any sufficiently powerful Italian sovereign might have given a thought or two to crowning himself "King of Lombardy" or even "King of Italy." However, the actual unification of Italy involved so many moving parts that the feat proved impossible until the aftermath of the Napoleonic conquest simplified things a bit.

Of course, it's impossible to provide an explanation as to why a historical event didn't happen. A historian can only examine what did happen. Small counterfactuals can be useful to examine individual events, but explaining why no individual or institution was able to unify Italy in the 14th century, or indeed any point between the tenth and nineteenth century, is another kind of thing altogether. I can, through some anecdotes, give some examples of things I think contributed to continued political fragmentation in early modern Italy, but I wouldn't call my explanations comprehensive. I've just gotten back on-line after two weeks off the grid, so do forgive my late answer and any long-windedness and feel free to ask any follow-up questions.

The first set of moving parts making unification difficult involved the sheer multitude of Italian states: every time it looks like one sovereign might unify Italy, push comes to shove and the largest states prove that they're actually too evenly matched for any one of them to make much headway. In the aforementioned example about Cangrande Della Scala, his state of Verona came to govern the cities of Belluno, Treviso, Vicenza, and Brescia, while Padua was subjected to a form of vassalage, and tight rapports of fealty were struck with Parma, Lucca, and Cividale. However, after Cangrande's death his nephew Mastino, no less an able ruler, found that Florence, Bologna, and Venice had struck an alliance to end his dynasty's hegemony in northeast Italy. Although initially the Scaliger Veronese state might have survived the conflict with their lands mostly intact (Bologna wasn't in any state to fight a war, Venice was only interested in culling further expansion, and Florence wouldn't realistically be able to project itself past the Apennines) Mastino was presented with a game-changing impasse when Lordoriso Visconti, escaped from his prison after having attempted to usurp the lordship of Milan from his uncle Azzone, sought refuge at his court. Although Lordoriso might have been a useful pawn in a bid to destabilize Lombardy had the Milanese and Veronese been on equal footing, with the Scaligers hounded by nearly all of northern Italy, Azzone was quick use his nephew's flight as a pretext to join the conflict (and machinated the blessing of the Pope, no less!)

Similarly, a generation later Gian Galeazzo couldn't proclaim himself King of Lombardy because he died of the plague and his less able successors saw the milanese duchy cut down to size by its rivals. And if fighting the rest of Italy proved to difficult, what if foreign powers were to get involved? Any claim to the title of King of Italy/Lombardy could easily provoke a war with the Kaiser in Germany, seeing as from the tenth century the crowns of Italy and Germany had been technically united in the Holy Roman Empire. Although Imperial authority was very much dependent on how willing and able the Kaiser was willing to descend into Italy to keep an eye on things, Italian autonomy had more than once unleashed long conflicts. It was much safer for Gian Galeazzo to just claim the title Duke of Milan, as it would be for any other ruler in Italy. In fact, one hundred years later Cosimo de' Medici felt it better to raise himself to the title of Grand Duke of Tuscany; one step below calling himself King.

But both France and Germany were particularly disinterested in Italy at the time of Gian Galeazzo's rule, and he certainly had a the largest state ever seen in Italy at his command. Would he have been able to proclaim himself king if he had, say, a decade more tacked onto his life and an adult son to help him? Maybe, the Duchy of Milan was certainly even more formidable than the state assembled by the Scaligers. But keep in mind there hadn't been a Kingdom of Italy four over for hundred years; there was absolutely no apparatus or precedent to support a monarchy in Italy. Would the Visconti have been able to build such an apparatus while also fighting a long war on multiple fronts? Probably not.

Why wouldn't they have been able to construct a pan-Italian apparatus of government? Well, another set of moving parts was was within the Italian states themselves: a very important factor which, a century later, would make it very difficult for France and Austria to occupy Venetian cities during the War of the League of Cambrai. You see, unlike, say, Germany, where the internal divisions were first incepted when Carolingian administrators entrenched themselves in the regional subdivisions they were sent to govern (more or less), in Italy the multitude of states were the result of individual dynasties subverting the collaborative form of government that had developed in Italy. The Italian system of governance was a great system to foster a complex economy that I think really can't be called "medieval" after the thirteenth century, but was in no way structured to centralize and scale resources the way a kingdom like, say France, could. Each Italian city possessed a uniquely complex governing council that balanced the representatives of urban workers, landed magnates, the clergy, and middle-class professionals. When single dynasties looked to subvert this system, they were engaged in complex balancing acts that didn't always allow them to make streamlined decisions (this is incidentally why cities like Verona and Milan were sometimes ruled by multiple members of the same family, and succession of "rule" can zig-zag from uncles to nephews and different branches of the same family). Oftentimes, the dynasties were more reliant on their own personal resources because the collaborative sort of governments in Italy often did not conceive institutions as separate from the convergence of individual citizens. Even in a uniquely centralized state like Milan, the Visconti dynasty's expansion was a two-way interaction between local governments and themselves, and overlordship could be easily toppled should local councils develop a powerful anti-Milanese faction. Viscontean rule over cities like Siena and Pisa, for example, was more a consequence of those cities asking for Milanese protection against their powerful neighbor (Florence) rather than a conquest born out of some grand design to unify Italy. Proving the ephemeral and conditional nature of this relationship, Gian Galeazzo's grand state evaporated on his death as his wife Caterina was left to govern the Duchy. The eastern provinces immediately revolted and after a brief resurgence of the Scaligers, defected to the Republic of Venice. As Gian Galeazzo's former soldiery revolted against Caterina, there was no more talk of projecting Visconten rule south of the River Po, let alone the Apennines, as Pisa and Siena sought to fend for themselves.

Follow up question:

How would the pope have fealt about it?