Can't see that possibly being competitive. For starters, the lack of standard, curated and supported ways of transfering FIAT would massively hinder volume.
Competition: What if every client could set their own desired fee? That's some serious Competition!
Lack of Standards: This would obviously need a core development team; I'd suggest modeling it exactly after bitcoin itself... In fact I'd love for the bitcoin developers to work on it closely.
Transferring Fiat: Volume would be whatever trades are within "quick enough" pingtimes. You're thinking about a central exchange here again... So once you use a broker to get your fiat up to your client in one form or another, the client just has to securely hold onto the credit there (I guess it would serve as your wallet too much like how mt.gox has a wallet for your bitcoins there) and trading is agreed between the buyer and the seller on a central client between them, not their own. (The client with the shortest pingtimes to both. This client's owner would get the fee.)
simple beats complex most of the time.
Because people were more stupid than they are being forced to become in this increasingly decentralized world.
P2P tech is the solution for every worthy problem known to mankind. World peace, economics, hunger, space research, every problem you can imagine is a problem because someone has power that is using it incorrectly. (A central point of failure.) Decentralizing takes power away from that point in every case and returns power to the masses on a distributed scale.
AnCaps like you and I will benefit from this trend the most, I'd think.
As long as "normal" exchanges continue to work, this isn't going to catch up.
Don't worry, I'm a free market kind of guy... I recognize that a newer, better exchange or three will come along to replace MtGox soon... And it's true that these exchanges will offer a trading environment with more volume than any decentralized solution could ever do...
...But there would be a strong set of reasons for the decentralized environment to exist, if not dominate. Most obvious among them:
1. Everyone is incentivized to run these clients in order to collect the fees. (Direct Income)
2. The rates on such a service cannot be affected by speculators nor DDos attacks, so it'll always be safer, and if nothing else a safehaven in times of stormy weather. (Such as when the centralized exchanges are being Ddossed again.)