I'm no expert in the field of ancient religion but I'd offer two reasons, at least for Rome. First, the priests in Rome were integrated into the political system (the "ladder of offices") and one could hold that office while holding other positions or leave that office for a better one, as Julius Caesar did, it wasn't considered a special "vocation" and it did not require a person to give up secular habits like getting married (and yes, I realize in the early church priests were married but there has always been a sense in which celibacy was privileged- so bishops were generally unmarried), so in a sense the priests of Rome were still as "lay" as anyone else. There were the Vestal Virgins in Rome and their celibacy comes close to distinguishing them from other people, but, as women, they were never politically competitive with other groups. Second, the clergy of the Christian church distinguished themselves from the laity through rituals. Only a consecrated bishop could consecrate other bishops and only a consecrated bishop or priest could confect the Eucharist. Lay-persons could not. Even when ascetics could challenge bishops politically and spiritually, it was still acknowledged that only consecration granted those specific powers. Paganism in general lacks this idea that consecration grants specific powers, and the priests in pagan religions were not the leaders of a community distinct from the political organization, as they were in Christianity until the Late Roman Empire, when that distinction began to blur and the first distinction was emphasized.

Those at least are my guesses. I keep think that the works by Peter Brown (who wrote at least one book with Gibbon and Hume in mind) would answer your questions better. Either "The Cult of Saints" or "The World of Late Antiquity" would be helpful. I have not read Robin Lane Fox's "Pagans and Christians: In the Mediterranean World from the Second Century AD to the Conversion of Constantine" but that also might be a place to look.