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I'm no fan of Greeley, but he is right in the sense that Trent attempted to regularize and clarify a lot of things that were, theretofore, rather "messy". While the educated churchmen generally avoided it, popular Medieval religious practice could be very eccentric, with the ill-educated peasants in isolated villages dreaming up all kinds of superstitions and novelties and mixing things up with old pagan beliefs. This was an ongoing problem that became more apparent as European society became more prosperous, urban and stable. Both the Protestant and Catholic Reformations tried to address it. The Catholic Reformation clarified a great deal of doctrine, provided more education for priests and attempted to squelch entrenched superstitions and abuses. The Protestants, and especially the Puritans, took everything overboard and ended up pillaging cathedrals and banning Christmas. Not in the Bible and so forth.
Greeley, of course, wants to somehow use this history to justify "progressive Catholicism" as a way to re-open stuffy "Tridentine Catholicism" to the organic, populistic "celebratory" Catholicism of the Middle Ages and Mexico, when it's about as far removed from that as it is from the doctrinal rigor of Trent. Just look at their "progressive" church architecture. Biege wood and utilitarian carpeting equals Puritan meeting hall, not the blessing of farm animals and kissing statues of the Virgin Mary.
Greely's also wrong, because Trent did affect Latin America. One of the things never acknowledged is that the Church had a huge number of indigenous American peoples to convert. The native population of South American was much higher than that of North America, and in addition, the Catholic approach was to attempt to approach and convert rather than to exterminate. Even though the Spanish government paid for missionaries, it was difficult to find enough to supply the needs in the Americas (including Florida, which was a Spanish possession and had enormous trouble getting priests to work with the Indians).
A lot of superstitious peasant practices were not entirely eliminated but were at least held at bay in the following centuries. I really think that the big problem was the lefty politicization of the Catholic Church after Vatican II. If you take away from people their good pious practices - the blessing of animals, prayers to St. Anne, the processions, etc. - they will be out looking for this level of religiousity in all the wrong places (santería, for example).
IMHO, Vatican II (or the "spirit" thereof) was 100% on the side of the Puritans; it was essentially an iconoclast movement, and it destroyed a Catholic culture that had been built up from the shared experience of many centuries, not just since Trent.
If you want to read a great essay, read Joseph Bottum's piece in the November First Things. I think it was entitled something like "Will the Swallows Come Back to Capistrano?" It dealt with the destruction of Catholic practice and culture. I don't agree with some of his analysis, but it is a great image.