In practice it's quite a bit more complex but that's the basic idea.
With PoW, it's still the case that the "rich get richer," because you have to be pretty wealthy in order to afford the equipment to mine any reasonable amount of coins.
But if I have no coins, I have to buy them from a coin holder with another form of money.
So this system would require the original issuer to be someone that you trust and have a reason to do business with in the first place right? So it's kind of like banks and central banks today?
I realize ICOs by startups are a thing but I kind of want to just ignore that whole phenomenon... I don't own any kind of coin and don't understand it well enough, but this seems to me to be a technology that will eventually be very useful.
They usually sell them in an initial offering. Of the top 8 coins on coinmarketcap only 4 have mining. Ripple, NEO, Stellar and Cardano don't.