I understand that religion is a wide subject, so I would like to stay focused on this one aspect for this discussion:
Also, please, let's just focus on the inner workings of one culturally coherent society. The numerous (and often bloody) consequences of a society meeting another with different one only right definition of morals is covered in many other sources.
My point is - in monotheistic (and if I understand it correctly to some extend in Hinduism too) religions, the deity is unified with whatever are considered the virtues of the society
The god in monotheistic religions is the (body-less) embodiment - or the very definition - of what is right.
The priestly caste then could draw their authority from the claim, that since they have higher knowledge of the deity, they can be arbiters of morals. It gives a guidance to asses whether a priest is a good one or bad and can serve as a basis for discussion about what role religion should have in politics and other questions.
In Greek and Roman religions, the gods are pretty much just humans (with both human virtues and failings) with superpowers. But how did their stories relate to anything going on down on Earth? And why would any power derived from them give you any authority over other people? What did Greek and Roman priests even do? And why?
LSofACO t1_ivva7ff wrote
There's a great series on practical polytheism here: https://acoup.blog/2019/10/25/collections-practical-polytheism-part-i-knowledge/
tl;dr the gods cause problems if you don't appease them, and sometimes even if you do. It's basically a reminder that before you do anything you should consider relevant factors that are outside your control: the weather, for instance.
EDIT: A more direct answer to your question is that the virtue being promoted by these religions is humility, or lack of hubris (a greek concept). They want you to never forget that forces outside your control and comprehension (personified by the gods) govern every aspect of your life.