The enlightenment and rise of humanism in the latter 1600's and 1700's attempted to shift this moral authority to "the people". And today, post-modernism attempts to put forth the notion that all morality is simply cultural context and relative. Which, while perhaps strictly true, is, IMO, pointless. Sort of like positing that we live in a simulation. Might be true, but so what? How does it matter?
Anyway, in today's world I don't think it's possible to have a widespread "trusted moral authority". Too many people seem to not realize the contradiction of saying on one hand that other cultures (and sub-cultures) should be respected while on the other hand decrying the utter horror of differing morals and ethics. Cultural differences are more than variations in language, cuisine, dress and music. Cultural differences are, at their roots, differing beliefs about what is right and wrong.
Kind of self-contradictory: the statement "it's all relative" is itself an absolute statement.
At least with respect to the US notion of limited governmental powers I think the goal was to keep the government from acquiring to much authority never mind something as all encompassing as "moral authority".
A moral authority implies leadership - that others would trust and defer to for moral judgment.
Isn't that effectively the government's job in a democracy? They're elected (directly or indirectly) to enact the will of the people. Unless you have a different scheme in mind for constituting this "moral authority".