The most overlooked place is at the hill opposing the acropolis: the pnyx.
That was the place where the democracy happened, where the citizens came together and vote.
The acropolis was like wall street and religion, the pnyx like Congress (forgive the American metaphor in a time where they chose the philosopher king over democracy /s )
But it being the birth place of democracy should be way better promoted, the Greeks should get back their Elgin marbles, but also erect a museum for democracy!