If you're asking what sort of institutional responsibilities the early modern political organizations in Italy entrusted to artists, the answer is very few (unless the artist was explicitly elected or appointed to a political post). However, art played an important role tied to prestige: setting and keeping up with artistic trends played an enormous role in political jockeying and legitimization of political personalities, acceptance of political decisions, and perception of political institutions. The pursuit and patronage of art could be political currency: a patron could increase his own social standing by introducing a skilled unknown artist to his political superiors, while a politician could earn goodwill by freeing up a renowned artist he patronized to accept commissions from a political peer either at home or abroad. A political appointee could use public art as a visible demonstration that they Get Things Done (it's harder to miss the new public building if it's covered in statues) while politicians can also use art in a more private sending to signal a change in balance of power (the council in the city of Siena, for example, commissioned allegorical paintings of "Good Government" to decorate the walls of their chambers after a political reform granted more power to representatives of the trade guilds). The ability to spend lavishly on art can signal to peers that your political, mercantile, or social career is taking off, and the kind of art you buy or commission as well as how you arrange it ("let's hold our meeting in the nice room with all the paintings") can transmit messages to the people who spend time with you.

Having said all that, the common narrative would have it that all the above materialized as a sudden phenomenon: it is thanks to the artists of the Italian Renaissance that patrons rediscovered and repopularized the art forms of their classical forbearers. While this is largely what the artists themselves would have you believe, the reality is more nuanced. No cultural moment exists in a vacuum: The quantity and quality (granted that the quality of art is always debatable) of Italian art had been on a slow upward trajectory since the eleventh century. And while war, plague, and economic crisis had intermittently interrupted the flow of resources available to the arts, between the mid-fourteenth century and the last years of the fifteenth century the Italian Peninsula experienced a period of relative peace and prosperity that allowed the arts to progress and flourish with particular rapidity and floridity. While war, disease, and crisis certainly didn't wholly spare Italy in this period, nor were all parts of Italy equally spared, overall conflicts were getting shorter and less destructive, wages and economic output was rising, and political upheavals and revolts were getting rarer.

Of course, the Italian leadership had a longstanding history of appreciating art. This appreciation transcended conflict, or lack thereof, on the peninsula: the thirteenth-century Divine Comedy, to cite a very prominent example, is a complex epic poem drawing on a multitude of references to ancient literature, theology, and modern politics. This poem was composed in the midst of (or, rather, largely as a response to) dramatic political conflicts that would see its author, Dante Alighieri, exiled from his home. Public art, especially decorative sculpture and architecture, further transcended conflict: when the late 19th century commentators derided the recently completed Milanese Cathedral as a porcupine-like neogothic excess, they ignored a tradition of elaborate decorative sculpture starting from at least the late twelfth century, when elaborate sculptures decorated the exterior of Milan's new gatehouses in defiance of the German Kaiser Frederick Barbarossa's descents into Italy to assert imperial control at spear-point.

But it's undeniable that even taking into account the comprehensive art history of Post-Roman Italy, the political leadership's relationship to art changed in the 15th century. The emergence of Florence as the leading polity in Tuscany certainly played a part: Florence was, until the Medicean usurpation (but through the cryptogovernment) administered by one of the most collaborative government systems in Italy. In Florence as nowhere else in Italy, informal networks of economic interdependence as well as social connections played an important role in the governmental policy-development process. In this environment, political figures were incentivized to develop cultured exterior images: this meant that not only was an upper class-Florentine's output of political and philosophical writing pretty an inverse function of time spent in political circles without receiving a political appointment (I'm exaggerating but not really) but patronage of the arts, and introduction of valid artists into a social circle, was a strong social signal. Raffaello Sanzio, better known in English as Raphael, was introduced to Florentine high society by a mid-level aristocrat named Taddeo Taddei, and would have probably remained a technically gifted albeit only moderately successful painter in Florence had Bramante, a more established Tuscan artist, not recommended his services to Pope Julius II in Rome.

Indeed, the migration of Florence-based artists like Raphael, Michelangelo, Bramante, and others from Florence to Rome in the early years of the 16th century was certainly tightly bound to broader political events in Italy: the Papacy was seeking to assert itself as a geopolitical player, collapsing republican institutions in Florence reduced the value of "prestige" investments for Tuscan aristocrats, and devastating new wars impacted the disposable income of patrons other centers of Italian Art like Milan and Venice. But in this migration, as with the introduction of artists to Florentine political circles, art flows from politics and not the other way around.

That is not to say artists never took explicit or implicitly political roles. Since we are talking about "Household Names," there is the very famous example of Lorenzo de Medici's presentation of Leonardo da Vinci to Ludovico Sforza. Lorenzo, of "Magnificence" fame, correctly perceived that Leonardo's active imagination and technical drafting skills would be appreciated in the militarist Milanese court of Ludovico Sforza. Out of Leonardo's relocation, Lorenzo and the Tuscan leadership obtained an innocuous excuse to correspond with the Milanese court outside of official channels, while the Duke of Milan gained the employ of a Tuscan-trained artist.