PoS is distributed not by random miners competing for a lottery, it is distributed by people (either directly or using an anonymization algorithm) that can choose to take (optionally anonymous) irrevocable capital control. Fiat is also distributed by people that maintain capital controls. So far there is no significant difference between paper money and PoS except that foreign agents can't counterfeit PoS.

I know that people are working on trying to make a fair PoS system, but it is turning out to be a wild goose chase.