>What did Greek and Roman priests even do?

Speaking as someone from a family that still practices polytheism, as most priests elsewhere, Greek and Roman priests were "privy" to certain divine rituals and rites that people wanted and demanded. This could be rites for good fortune or luck, favour from the deities, protection against harm or evil, or even curses against rivals and foes.

For many of these services, fees will be charged. A temple or sect might also receive patronage and donations from wealthy or powerful believers who want to show their piety or devotion to a deity (or at least the public perception of them).

>In Greek and Roman religions, the gods are pretty much just humans

That just simplify things for people that practice these religions. If the gods are just humans with super powers, then the rational is that they will behave and act like humans. My neighbor Marcus might not be a saint, but if I gift him an jar of wine, he'll most likely be pleased and do me a good favour in return. Likewise, if I sacrifice an amphora of wine to Jupiter or Juno, they'll likely be pleased by it and answer my prayers. In Roman religion, this concept is known as do ut des ("I give that you might give").

Keep in mind, morality is subjective. Just because your version of morality doesn't wholly applied to the beliefs and mythology of Greco-Roman polytheism, does not mean they didn't have their own form of morality in place. What you perceive as "good" or "bad" might be different for a Roman or Greek living two thousand years ago.

>And why would any power derived from them give you any authority over other people?

I mean, you can say the same about monotheistic religions. What gives the Catholic or Islamic priests the authority to represent their monotheistic god on earth? Many instances, monotheistic religious figures have acted and behaved contrary to their "moral" teachings of their faith, yet their members still followed them. These monotheistic institutions have also been challenged and supplemented throughout their history, so they're not exactly invulnerable to strife and discord.

Traditions are powerful thing. Once a religion - mono or poly - establishes itself in a local society, it tends to proactively maintain its place in a way that keeps the people believing in the system.