How could Bitcoin-style technology power a global fast peer-to-peer payments system with minimal fees without becoming its own currency?

You could create a peer-to-peer (trading) payment system (with some added trust), the likes of which CurrencyFair or Transferwise are doing. The catch would be traction. You would need a lot of users for the payment network to really take off (look at what Ripple is trying to do - see Ripple) - which is essentially a p2p payment network, on the basis of the question you have asked.

Technology and adaptation is not the issue. The issue is of going to jail. The problem is, financial regulators in each country do not like this, and will try to scuttle the network in play as well as the players.

There are many companies and startups that are actively trying to find a legal method in trying to deploy peer-to-peer payment networks, so that trading and making payments on a global scale comes as easy and cheap as a Skype call.

Have no doubt, the existing banking industry and the financial regulators would not like to see this happen.

There is no doubt, P2P payments will take off, but the owners would be the likes of Bank of America, JP Morgan Chase, etc. rather than some decentralized network (unless of course, Ripple really takes off!)