I don't think so, because: BTC Banks cannot borrow Bitcoin from a central bank to buy more time.
True, no last lender means they have a bankrun risk, so that they should treat each loan much more carefully. This is by design
Exactly. If loans can still theoretically be made and fractional reserve banking and interest can still be part of the loaning process, the fact that there is no last lender and the fact that it is a well know fact that there is a known total maximum amount of coins make banks run risks very high.
It makes them higher than what you're used to, that does not make such risk "high". A bitcoin bank that fractionally loans out twice as much as it keeps in reserves can crediblely do so, so long as it's open and up front abou it's methods and long term CD's are it's primary method of raising capital (or some other not-on-demand deposit system).
Except that the bank can't lend out bitcoins, but only something that can be redeemed for bitcoins. Would you accept a piece of paper that says: "can be redeemed for 1 Bitcoin at the first international bank of bitcoin at any time" instead of 1 Bitcoin?
You've got it backwards. The borrower gets actual real bitcoins. It is the depositor that gets the IOU.