Thousands of private currencies exist right now.
Nobody would care about your currency, since it is useless, and as such worthless. A centralized currency is an enormous social asset. Nobody wants private currencies.
>who could be trusted to do it in a way that doesn't favor some at the expense of others?
Just try to imagine a currency controlled by amazon. Do you think they won't do some hyperinflation making you poor once a decade?
The controller of a currency has enormous power. If the government can't handle it nobody can.