For centuries the Church was the cornerstone and guardian of western morality, even though it sucked at it then as well. So its legitimacy was established by fiat, and challenged in the late middle ages when enough middle class non-monastics were taught to read and could form educated opinions.

That said, since we have a long and well known recent history of the Roman Catholic Church showing questionable judgement across many matters (and all other religious institutions doing no better) the public doesn't pay them much mind, though through lobbying, they have a lot of influence over the government.

To be sure, if there ever is another Reign of Terror, much of the cloth will be in the guillotine line.