So in hell we solve this fundamental problem? Can Lightning Network channels in the future be used to assemble proper exchange replacements so I no longer have to sleep scared that the next MgGox/Mintpal/Cryptsy/Bitfinex/... is going to happen and I'll lose what I keep on the exchanges? (a small % compared to my total holdings of course, but still very annoying).
Lightning Network channels could be usefull for decentralized crypto coin exchange, but I dont think you can actually trade fiat in trustless manner (real time and in huge volumes). Thats the problem with fiat, it has no public ledger so there is no way to prove in trustless manner fiat tansaction happened. But then, just decentralized crypto coin exchange is not much interesting, more important for people is actually fiat <=> Bitcoin exchange.