You ask about huge time period, nearly millennium, and much changed within that time frame, so, as in many cases, there is no short and simple answer. Closest thing to this would be to say that theres always was someone in Papacy's way:

At first it was Charlemagne's creation- empire. Papacy had some success maneuvering among powers during the fragmentation in late 9th- early 10th centuries, but rise of Otto I sealed any chances of Papacy's territorial expansion. South Italy was united (with Papal blessing) by Normands and polity they established- Kingdom of Sicily- persisted until 1870, again halting any chances of advance in this direction. Only after fall of Hohenstaufen, Empire stopped being a factor in Italy, but Papacy got involved in conflict with France that escalated so much, that in 1309, pope backed by France refused to enter Rome and settled in France, city of Avignon, effectively removing Papal states as power player. Latter the was the Great Schism and Reformation, then Papacy (and all Catholic world) had bigger and more pressing problems and since Renaissance Italy was battle ground of bigger players- primary France and Hapsburg's- leaving little place for expansionism of any Italian state.

Also, Papacy is an elective theocracy. Henry II or Richard I inherited they power base, enemies and claims in France, while every pope come from different background or another country. Let's say in 12th century, the peak of Papacy's power we have: commoner with rural background (Honorius II), commoner with urban background (Eugene III), an Englishmen who spend most of his life in monasteries in France (Adrian IV) and a bunch of offspring of various Italian noble families. With such diverse shifts in leadership, Papacy hardly could had a long standing territorial policy.

That brings to another point- Papacy had continuity in politics, simply unifying the Italy was never the goal. Popes considered themselves leaders of entire Christendom and pre-occupied themselves with larger issues that securing fealty of Siena (The Donation of Pepin itself happened first of because Lombards were trying to subdue Rome itself and demanded submission from the pope). Some popes were involved in temporal politics very much, but for most part, Papacy acted as power above dukes and kings.

Essentially, even if we assume popes wanted this, there always was bigger power in the way.

Oh but they did expand, at their greatest extent the Papal States were the second-largest Italian State by land area. Any study of Italian history will reveal that the Papal States played a central, often decisive, role in post-Roman Italian history all the way up to their absorption into the nascent Kingdom of Italy in 1871.

Part one: The birth of the Papal State

In the seventh century Italy was a battleground between the Lombard warrior-aristocracy entrenched in the Northern Italian fortress of Pavia and what was left of the Roman Empire; centered on Constantinople. The Empire of the East ruled Italy from Ravenna, the historic base of the Adriatic Fleet protected from the mainland by swampy shoals. The Governor of Ravenna, the Magister Militum ("Military Magistrate") appointed Dux to govern the areas still held by the Empire. These were almost exclusively coastal areas, often urbanized: the Pentapolis in the northeast, Calabria and Apulia, Naples, and Rome. The only exception was Perugia, an important fortified town that commanded the narrow land-corridor connecting the City of Rome to the Roman heartland centered on the Pentapolis.

Although the Apennine corridor kept open by Perugia did somewhat cut off the Southern Lombard Duchy of Benevento, the corridor was more than permeable: it was in constant jeopardy. Consequentially in the City of Rome imperial authority was often distant. The Pope, or Bishop of Rome, per Roman administrative practices was a functionary of state and accrued onto himself the powers normally reserved for the Byzantine Dux. Late-fifth/early-sixth century Pope Gregory the Great (590-604) took many steps to establish how the Bishop of Rome would run the city, incidentally also reducing the importance of the Senate, which believe it or not still existed.

The recognition of what had up until then been the Byzantine Duchy of Rome as an independent state was a product of expedience, coincidence, and possibly an afterthought. You see, in the second decade of the eighth century, King Liutprand of the Lombards, probably more to have a direct line to his unruly vassals the Duke of Spoleto and the Duke of Benevento rather than any grand scheme to conquer all of Italy, moved decisively to occupy strategic castles in the Byzantine Corridor connecting Ravenna and Rome. Pope Gregory II, however, was seriously worried that Liutprand was moving to remove remaining Roman authority from the peninsula. So Gregory opened negotiations with Liutprand to return the occupied fortresses to the Exarch of Ravenna. Liutprand, who had rapidly whooped his vassals in Benevento and Spoleto, evidently said something along the lines of "Meh" when faced with the Pope's proposal, gave the Pope the fortified town of Sutri, and went on his way.

The donation of Sutri sets a precedent: The Pope clearly isn't a Byzantine functionary in Italy anymore. The Pope is one thing, and the Byzantine Empire is another thing.

When Liutprand's nephew Astulf was elevated to the throne in 749 after the usual succession kerfuffle, the Lombard position in Italy is very strong indeed. By 751, Astulf has not only laid siege to Ravenna, he has taken the city. Pope Stephen II, initially charged with carrying peace negotiations on behalf of the Byzantine Emperor, decided instead to throw his lot in with King Pepin of the Franks and traveled to the Frankish court, at the time in Pepin's summer manor at Quierzy.

Astulf was rampaging up and down Italy squashing all the Byzantines he could find, a strong position indeed. However Pepin was much stronger: the Frankish heartlands were more secure than the contested landscape of Italy, and while the only thing that occupied the Lombard aristocracy more than fighting the Byzantines was fighting amongst each other, Pepin had a much more stable power base (sure his reign saw small rebellions, but they were squashed with minimum fuss). In exchange for Papal legitimization of his rule over the Franks, in 754 Pepin descended into Italy and defeated the Lombards. In spite of loud Byzantine protests, the Papacy was handed an expanded reign over the former Imperial lands in Romagna (the last Roman strongholds; in other words the northeastern Pentapolis and Ravenna) in addition to the Perugia corridor, all under the protection of the Frankish Empire lest the Lombards get any new ideas. Pepin was even made a Roman Patrician, a title which until then only the Roman Emperor had conferred.

However, at this point in time, there was no structure in place for the Pope to rule the new territory. Local administrations, often centered around bishops or parishes, were entrusted to govern. Subsequently, asserting Rome's authority within the borders of the Papal State would be a recurring theme in the State's history.