They provide services. What does a cashier "produce" at the grocery store? There were also artisans of all sort that did produce finished or intermediary goods - brickmakers, pottery makers, blacksmiths, weavers, animal breeders, and on and on and on.
the athenian system, as per the UVM article, doesn't align with modern capitalism. It had democratic features like paid public service roles, but was primarily an oligarchy, not a free market. economic participation was broadened, but the structure remained class-based, with power concentrated among the elite.
It's economy wasn't modern free market system focused on capital accumulation and investment for profit. It was a mixed economy with significant state involvement and a variety of revenue sources that went beyond simple market transactions.
State actors paying their civil servants isn't evidence of capitalism, or wage labor.
( David Graeber does write about ancient Greek city-states and how their coinage came to be, according to the historical and archeological record, in "Debt: The First 5000 Years," by the way. )
Capitalism, is a relatively modern phenomenon, with a pretty common, well understood definition. This isn't a heretical or radical idea. You may find some societies prior to the 16th/17th century that fulfills some characteristics of capitalism, but they don't make the cut.
People naturally want to trade. Markets existed before capitalism and they will exist after capitalism.