Traditionally it was the established/predominant religion's job, not the government. It was hoped by many that religious institutions could act as a counterweight to the nobles, kings and politicians. Which is why the civil power structure tried (often successfully) to co-opt religion as well. Of course, religious institutions have their own issues in that it often becomes a parallel power structure on its own. Or go from reflecting cultural norms to shaping them.
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.