Such bitcoin banks can -prove- they have at least that much cash on hand, and thus can prove their compliance with laws requiring that their deposit must be over their loans by at leat 10% for example.
Aaaand, you've gone off the rails again. You made a great case for deposit banking and loan banking. What on earth does this have to do with fractional reserve banking?
Maybe I fail to understand something (I'm not an economist, I'm a physicist), but what I understood from my introduction to economics class in high school was that deposit/loan banking = fractionnal reserve banking.
If I go to a bank to deposit my money, and then the bank lends that money to somebody else, then two people have some "claim" to that money. Of course, the lended does not "own" the money; he'll have to reimburse later, but he has it on hand, and may use it do do economic activity. On my side, by having lent money to the bank, I may legitimately claim I own that much money, especially if there's is a guarantee on my deposit, which I believe is the case in all western country for checking/saving accounts.
Of course, my money in that bank is not real cold hard cash (they don't have it, they lent it), but because it is guaranteed by governement, it has almost the same value. (To most people at least) The bank IOU for one dollar has the same value that a one dollar coin. I can transfer that IOU to somebody else (exactly what happens when I write a cheque, or use my debit card), in exchange for a good or a service. These are the same "virtual BTC" others have talked about.
Thus, for a given "amount" of $ deposited in the bank, twice as much economic activity has happened. Which is exactly what happens in a fractionnal banking reserve.
I guess the only way to prevent that would be that when people deposit a certain amount in a bank, it should always be locked for a number of years/equivalent to the duration the bank will lend it, so that it can never finds itself in a situation of a bank run, where people withdraw their money faster that loans are repayed. (But I'm not quite sure how this would work with bad loans). But I'm quite sure that even in a world ran by BTC, people would still like to have their money available at all time, even if this implies slightly higher risk/lower intersts, especially if the deposits are guaranteed by the government.