It would seem the "blank" you described would be the Lombard Kingdom of Italy, the Carolingian Kingdom of Italy, and the union of the German and Italian crowns under the Ottonian Dynasty.

Not that this shouldn't discourage additional discussion, but I've written about this topic multiple times in the past, and you might be interested in these older answers of mine: I explained the main drivers behind Italian fragmentation in this answer, but if you're interested in more minutiae I also wrote more specific descriptions of the ninth, tenth and eleventh centuries, diving into some of the driving factors behind the emergence of Italian urban republics. I also wrote this shorter but broader answer explaining some of the more governance and government-oriented trends.

In any case, the short answer is that it is not a consequence of happenstance that the center of Europe's greatest empire was a peninsula characterized by dense urbanization, high agricultural productivity, and strong internal and external trade links. This afforded the two post-Roman kingdoms of Italy (both Lombard and Carolingian) an enormous potential for internal instability: with relatively small power bases, aristocrats could easily challenge central authority; while figures of authority, such as they existed, even in times of tranquility could never adequately concentrate authority. Even under the Frankish Empire, imperial representatives and margraves alike had to contend with clergymen, judges, and lawmakers seated in the cities who were elected, in the Roman fashion, by colleges of prominent citizens. Indeed, thousands of large and medium sized landowners split their time between country estates and urban residences, while small farmers, aided by the shallow waterways crossing the plain and a weak legal precedent for obligations to their landlord, regularly traveled to town in order to bring their goods to market and purchase finished goods made by urban artisans, all contributing to the liveliness and bustle of urban life.

The urban-centric power structures fostered inclusiveness, enfranchisement, rule of law, and as a consequence, trade and prosperity. A collaborative sort of government emerged, whereby leading dynasties ruled together in "Communes." In fact, the sort of landscape you described in your original posts only starts emerging in the thirteenth century, and really kicks in gear in the fourteenth. It is only then that there is a true ossification and consolidation of the Italian political landscape. Prior to that point the urban "Communal" governments in Italy ruled over a dense network of interconnected urban, suburban, and rural territories. They are in no way comparable to the German or French duchies of the time; and are better described as city-centric amalgams bringing together urban artisans, landed aristocrats, and an entrepreneurial middle class that could be a combination of both.