Much of Michelangelo's work was produced for the House of Medici, in large part to create a sense of culture around the newly-powerful clan. And while, yes, the family produced four Popes, though more as an outgrowth of their political and financial power consolidation than strict religious interest.
Again: the Church was very different during the period than it is today, representing political power, a center and repository of education and knowledge, tremendous financial power, and a very active role in political goings-on. Again: it's really not particularly surprising that the Church was so involved in sponsorship for the arts.
Today those roles are largely filled by commercial interests, an insight which first came to me in the late 1980s as an early computer animation festival was touring through my college town, and I realized that many of the best shorts, almost all advertisements, were for tobacco and alcohol companies. Today that niche is filled more by the FIRE sector.