There is little to no incentive to manipulate the system, as doing so would be immediately verified and determined by the auditing system (blockchain of BCT being out of whack with the vaulted $FIAT reserves). It would be like saying Mt. Gox would just up and close shop and take all your fiat and bitcoin tomorrow. Not that they couldn't, but why? Why kill the golden goose (fee structure)? They would have the worst reputation on the planet and could potentially kill the bitcoin network overnight. 90% of the current miners would bail. That just doesn't compute.
Most (vast majority) of the people don't keep their money in MtGox. If MtGox were to run off with the funds, at most they would have a few $100k to play with. They have more of an incentive to hold as little as possible (reduces their liability), and to keep making money off trades, though even that isn't guaranteed to be a viable long term income.
In your system, who is checking the physical fiat reserves? Why should we trust that they are honest and not working with vault holders to skim off the top? And why would they care about reputation of they have the option to run of with ALL the Fiat, as opposed to just a fraction of it?
We are already in this situation with the top 4 $FIAT/BTC exchanges, otherwise you would just trade your bitcoins with another bitcoin address and never mess with the exchanges in the first place. If you want fiat we already have a very centralized "fiat-backed" system. The majority of players in the market are probably storing both their fiat and their bitcoin with Mt. Gox (80% volume leader).
As I said, I seriously doubt a lot of people are holding their money in MtGox. Especially after all the losses and scandals last summer. High volume just means two people exchanged money using their system, and likely withdrew it as soon as possible. It doesn't mean MtGox sold BTC themselves and are keeping the money, as would be in your scenario. The point is not how much, or who, but how much control is there by free individuals, and how much of our well being depends on having to trust someone else.
The $21B fiat just sits in the vault. It goes nowhere until someone converts BCT for fiat. If everybody bails, then the BTC/FIAT pair value will change, exactly as it does now. Think of it like shares in a company on the stock market. If you have more sellers than buyers the stock drops, conversely for more buyers than sellers it goes up.
Ah, I think I see where you're going with this. You want to essentially create shares of stock equivalent to $21m or whatever, and sell it on the market with the promise to keep the share exchanging system secure, and the price pegged to $1 per share. But, again, corporations issue and sell shares. For the promise of giving profits and being secure, they ask that you give them lots of your own money, which they then use for whatever they want with the condition that the buyers of that stock will like what is they do. Again, how is your proposal different from also asking for money from people that you'll be free to do with as you please? If it's not, I guess your main issue would be trying to sell your idea so people can give you their money.