In the equilibrium, the average consumer buys from the average firm. Otherwise it doesn't stay average for long.

The median consumer of a good purchases from (somewhere around) the median firm selling a good. That doesn't necessarily aggregate, and it certainly doesn't weigh all consumers or firms equally. The consumers who buy the most of a good tend to have different preferences and research opportunities than average consumers, for example.

You could get similar results in a democracy, but most democracies don't really encourage it : most places emphasize voting regardless of knowledge of a topic, and some jurisdictions mandate it.