I do not know if the Florentine Republic was more or less volatile than similarly-sized Italian city-republics. Starting from the thirteenth century, Bergamo, Verona, Padua, Milan, Cremona, Bologna, Siena, and Genoa were forced to contend with organized military actions commanded by their own exiles. Perhaps Florence had it easy!

Certainly, the governmental model which propped up the Italian Republics was a complicated amalgam of groups with very different interests. The urban clergy, the large and small landowners, the middle-class professionals, the artisans, and of course the working poor, all interacted in myriad and sometimes violent ways. Some authors have argued that the prospect of these groups peacefully collaborating in government was altogether hopeless. Politics in Italy was not for the faint of heart: exile, loss of property, and loss of life could be the consequences of finding oneself in the political opposition. But commonly, when two feuding political parties sat down to compromise the only outcome was the realization that there was no compromise to be had.

So perhaps volatility is indeed the exceptional quality of medieval Italian politics. The Italian urban communes were strange, ad-hoc polities uniting different legal jurisdictions, prerogatives, and public will. Their own founders were weary of their creation, and most outright refused to legally acknowledge their own sovereignty: in medieval Italian jurisprudence, the Empire continued to be sovereign, albeit in an increasingly nebulous capacity. In such a climate, is it surprising that the smallest of tensions could erupt into serious conflicts? Some authors affirm that the communal age was itself one of crisis: there was no overall authority in Italy, and the communes were a response to this; a haphazard stopgap at best. There was no state to protect opportunities and civil rights, leading different groups of people as disparate as military companies, associations of noblemen and commoners, merchant guilds, and the guilds of small shopkeepers and artisans, to all come together in tenuous systems of government. The loyalties of these people lay first and foremost with their order within society, guild, family or dynasty; what have you. For in spite of the communal superstructure, it is a recurring theme within medieval Italian politics that it was these associations which defended people and gave them a place at the political table, holding the power to improve their lot in life. There is strength in numbers, and birds of a feather flock together.

It's also true that crises are history's punctuation marks. No one writes epic poems about how content they are about everyday life, you don't become Florence's greatest banker without making a few enemies, and you don't write one of the world's most enduring political handbooks unless you've really seen things. The communes were just as driven by intense economic activity as they were by Italy's heritage as the cradle the Catholic Church and the Roman Empire, and economic activity cannot flourish in a climate of endemic instability. Most of the time, life in the communes was stable and prosperous. The communes might be subject to internal threats orchestrated by disgruntled parties, but more often than not the greater whole was able to defend against attempts at usurpation and upheaval. Even when single groups or dynasties were able to come to power in spite of these communal institutions, this was almost always via the exploitation or placation of the governing apparatus, not its dismemberment.

There is, I suppose, a trend which takes place sometime in the 13th century, but that's earlier than the focus of your question. In short, the communes were almost too successful for their own good, and the growth of prosperity and acceleration of economic activity led the lower middle class to rise up, clamor for, and almost always obtain political enfranchisement to the detriment of the ancient urban dynasties and their rich vassals, many of whom still possessed feudal rights outside the city walls (feudalism didn't exist, we are all wont to snidely repeat; but its single parts very much did). In most cities, the process almost always took close to a half-century, and was characterized by bursts of intense conflict.

Florence, perhaps, because it retains its republican institutions so late into the 15th century, might appear to have a particularly convoluted political history. Or maybe because of the disproportionate influence of Florentines, be they artists or politicians, we are prone to think that Florentine politics was particularly unstable; even though other communes in Italy exiled out out of favor figures in equal proportion. Certainly, Florence was the most successful polity in Tuscany, and one of the more successful in Italy. That in and of itself suggests that its political system must have been doing something right. I don't know if I answered the question to your liking; feel free to ask follow-ups!

Michelangelo was exiled from Florence? This is news to me. He certainly spent long, long periods of time away from his home base of Tuscany, most of his life was spent in Rome working on various commissions for the Church, if I'm not mistaken. But as far as I know he was never "exiled". Unless you are referring to his dutiful following of the Medici family during the early years of his life after the Medici proper are ousted for Piero the Unfortunate's failings concerning the French invasion. Even then, he returned to Florence relatively soon afterwards, finding little success with his still-nascent career in Bologna. Ironically, he did so under the protection of a Medici family cadet member by the name of Lorenzo di Piero Francesco di Medici as at the time Florence had fallen under the spell of a self-assumed prophet who had transformed the city-state into a virtual thoecracy. Even if you did count following his patrons and familial benefactors out of Florence as "exile" that would fall under umbrella of Piero the Unfortunate's exile, wouldn't it?

Or perhaps you are referring to his flight from Florence after he backed the ten-year long rebirth, then untimely death of the Florentine Republic after Pope Clement VII (who was a Medici) retook the two of their's mother city (they were childhood friends, did you know). Again, exile might be his de facto status, but as far as I know he was not de jure exiled during that time, and in fact was enthusiactically invited to return to Florence by the ruling Medici at his time during the last segment of the sculptor's life in Rome. From what I understand, Michelangelo ran afoul of the Duke of Florence at the time and decided to retreat to Rome with the Pope's blessing.

Now, Michelangelo may have considered and conducted himself like an exile from his beloved Florence as a bit of a dramatic metaphor for his work, but as far as I know, at no point during his entire life was he ever not welcome to return home, whether Florence was as a Republic or as a puppet state under the Medici.