You are nitpicking.
From Wikipedia: "The Pisa Baptistery of St. John: Construction started in 1152 to replace an older baptistery, and when it was completed in 1363, it became the second building, in chronological order, in the Piazza dei Miracoli, near the Duomo di Pisa and the cathedral's free-standing campanile, the famous Leaning Tower of Pisa."
On the other hand: "The Black Death arrived in Italy by sea, first making landfall in Sicily in early October, 1347. By January 1348 it had landed in Venice and Genoa. A few weeks later it appeared in Pisa and from this foothold it moved rapidly inland, east through Tuscany and south to Rome. By the time it died down in the winter of 1348 more than a third of Italy’s population had perished." source
Whether anyone became rich is mostly irrelevant, I chose to focus on the outcome, the cathedrals themselves, which are still here several centuries later.
And here too, you are mistaken: by enhancing their status, the cathedrals greatly enriched the priests, cardinals and more generally the Catholic Church (though not the artisans). Someone did became immensely rich, just not the ones most people were looking at.
Anyway, you better start blogging rather than merely criticizing :-P