In the Medieval Italian context (and, I always assumed, in the Ancient Greece as well, but I actually couldn't say for sure) the phrase "city-state" is a bit misleading: the state's authority did not end at the city limits, and indeed often extended far beyond them. In fact, in Medieval Italian political jurisprudence there is a weird lack of territoriality: typically an Italian city-state government, called a "commune," defined itself exclusively as a group of people coming together and subjecting themselves to a greater authority. That authority need not be concentrated in a city. Although it's admittedly difficult to envision a non-urban situation where you could construct this kind of government, there is one very prominent example: Venice. For much of its early history, Venice was actually a collection of islets divided by thousands of square feet of brackish water. You can read about medieval Venetian food security in this old answer of mine.

It's also worth investigating the logical corollary to your assumptions: if the city-states didn't govern the countryside, who did? And why would the city authorities even negotiate with farmers instead of simply having them come in and sell their food in a marketplace?

Almost every city prior to modern communications and transport technology could only exist exclusively thanks to an intense and continuous dialogue with the surrounding countryside (although I learned about a notable exception earlier today: Imperial Rome). In the context of Medieval Italy, one of the very first social groups to embrace the Communal mode of government were influential landowners, interested in ensuring their tenants could safely and profitably get their goods to marketplaces (and pay their rents!). In fact, the Communal political system was characterized by periodic conflicts of interests between the "Consular Aristocracy," that is to say the important urban dynasties who often also held large estates and titles; the "Suburban Nobility" who were either vassals of the large urban dynasties or free landholders; the middle class professionals, the most successful of whom could also own estates outside the city; the artisans; and the working poor. I actually talked about these different groups, and the ways they could come into conflict, in this post just last week.

Although the Mediterranean has a long and well-documented history of urbanization, one of the reasons why the city-state form of government was so successful in Italy is because of permeable barriers between segments of society, which also included a high degree of mobility between town and country. Although Medieval Italy was, like the rest of Europe, a frighteningly unequal place, the fusion of the upper tiers of the bourgeoisie with the lower tiers of the nobility led to a varied and nuanced interplay between town and country: there could be cases of members of the upper middle class like notaries, lawyers, bankers, and moneychangers imitating the aristocracy, and own estates outside the cities where they lived. There were also some aristocrats that spent vast amounts of time in their city homes while neglecting their estates, while some members of the upper bourgeoisie who owned comfortable estates might only go into the city once or twice a week. But of course there certainly were other factors too: importantly, Italy's dense urbanization and multitude of cities is a holdover as the peninsula's history at the center of the Roman Empire, but this very urbanization is certainly facilitated by a very fertile landscape. Factors unique to individual cities, like Milan's complex hydrography, contributed to suburbanization and the spread of the city's authority far past the city walls. You can read more about that here.

In sum, we shouldn't get too caught-up in the notion of "City-States" in Italy. Rather than considering them states which consist of a single city, it is best to think of Medieval Italian polities as states disproportionately centered a single city. Within the jurisdiction of these cities were countless suburbs, country estates, farms, and sometimes even other cities! By the time these states fused into the Italian States of the Renaissance, it is most definetely inaccurate to call these polities city-states.