In a very simplified way, proof of work is a way to roughly uniformily distribute the block generation "queue". In a network without identities how do you prevent someone creating hundreds of accounts to have a hundred times more chance to be next in line? You give them a hard problem, and now the odds of you being next in line is proportional to your computing power, which you can't multiply effortlessly.
Proof of Stake says that instead of distributing the work queue proportionally to the computing power you demonstrated to have, it does it proportionally to the amount of currency you have saved. It similarly prevents the attack where one could create infinite personalities to get in line, with different trade offs. In particular, beside the energy savings, it can be a much more scalable model, where you don't have to wait 10min in average for someone to solve the hard problem and instead you can know right away who are the next people eligible to generate the next blocks.
You can look at the block rewards and mining fees as interest on your investment. The "saved" part is interesting too - I've seen variations where you have to lock your stake for a longer period (trading rewards for liquidity) or where you have to "burn" an amount of stake, permanently taking it out of circulation to get a spot on the line.
The inflation rate and actual utility of the currency are important factors too - if most of the currency is already distributed and the inflation is low (thus block rewards are low), the richer don't get much richer. If the utility is high, redistribution occurs more naturally as well.
Perhaps the inflation is not low but there are other distribution mechanisms that distribute currency based on utility in a higher rate than block rewards (for example on steem, of all newly minted coins in a block ~5% goes to the block creator, ~65% goes to content creators, ~6% to commenters, ~17% to curators and 7% as interest to those that have commited their stake in a long term deposit).
But yeah, in the end it indeed is a factor which is one of trade offs I mentioned, but there are ways to combat it.
For block rewards you'd have to mine. In particular you'd have to have a server running a node joined in the network ready to generate a block whenever your number is called.
Not everyone would want to do that, so there's usually a mechanism to indicate you want to be eligible to do so, depositing your coins or similar.
That's for block rewards - nothing prevents a blockchain currency to be designed to generate inflation and add interest on everyone's investment - some do exactly that.
If it works as described, yes, but no more so than any other investment - and if the interest/profit on those coins are higher it will draw capital from other places, which will decrease the interest/profit on the coins.
The problem is in the details: how do you use this to reach consensus? How do you agree on a randomness source to picks the "winners"? How do you prevent people from mining multiple parallel histories, devolving the system into proof of work?
Because PoW enabled trustless transactions AND a provably equitable distribution of the coin via distributed minting. "I can prove that this is not a scam to waste your time and money."
Other non-mined coins would never have taken off because they would never have gathered the critical confidence/interest that Bitcoin did. Only after PoW paved the way could others have any faith in DPoS or other coins.
