We have to distinguish different use cases a blockchain technology is adequate for:
The whole crypto currency technology is based on a consensus protocol (all the nodes do agree on a ledger). Obviously, it can lead to both public and private chains to keep track of transactions.
Traditional banks can use it as a core banking transaction processing system instead of their current saving/retrieval infrastructure which is highly vulnerable to single point of failure and forgery threats, (there exists a lot of projects in different stages with this application in mind) and there will be no need to publish a new currency (just use the same unit of price, conveniently). In this scenario, privately owned, 'somewhat' trusted nodes, reach to the consensus state very easily and in a cheap and fast way.
For the public chains, the consensus protocol needs to be costly to participate (otherwise what stops hackers to issue malicious statements about what is legit or not?) and so you have to be prepared for a reward system to redeem the costs and even more. This is how we end to genesis, 'creating' and publishing a new money (BTC, ETH, XMR, ... ). Eliminating genesis, (when you want a government backed currency, you should forget about genesis, shouldn't you?) forces us to increase the transaction fees, which is not in favor of the people who decide about the transaction processing infrastructure of choice, ultimately.
But, as guys have mentioned earlier, we have success stories for one or two pegged currencies running on the top of public blockchains and do not publish a new coin by genesis mechanism and stick to the fiat currencies, these are 'pegged' currencies (pegged to USD , EUR, ... ).
For instance, Teher is doing well right now, take a look at their white paper here:
https://tether.to/wp-content/uploads/2016/06/TetherWhitePaper.pdf
