This post just follows up from the last where I compared a bank to a supermarket which retains it's earnings. There is an important difference between the two, which helps to clarify the point that has been made.
As far as I can see, the difference between an earnings retained Sainsburys (if excess profit-as-cash isn't given back to shareholders but retained, say in a vault) and the desert-island bank - is that Sainsburys sucks in cash; whereas the desert island bank has an outstanding /obligation to receive cash/ which may or not be covered by any 'real' cash existing in the system (it is conceivable that there is not enough cash & deposits in the system to substitute for cash). Bank deposits /can /substitute for cash (until all the deposits are used up) but in this thought-experiment world, there can be a retained obligation for the guy to supply something (cash) which he can't get from anywhere since it does not exist outside the banks and the central banks. In that case there will be large amount of unrequited demand for cash (the debt obligation).
There appears to be only a few ways that cash can be got into the non-bank system, to supply the outstanding obligation. The main options are if the cash in the bank or central bank is used to buy up real assets. Presumably if the bank(-owners) have market power then they would be able to enforce as low cash-price for the assets in exchange for the outstanding debts (in other words there will be a fall in asset prices). In other words, the removal of the outstanding debt obligation /does not have to be /in the form of a default; it could be that the bank invests in property directly; or that the earnings are distributed to shareholders, who themselves invest in property. However, the bankruptcy choice is /more likely/ if asset values fall (and if the decision rest primarily with the debtor) because bankruptcy transforms the obligation to hand over the now-low-value asset rather than a costly cash-obligation.
Thus deleveraging will cause a fall in asset values and either
a) a settlement of the outstanding debt due to the banks or the shareholders of the bank physically buying real assets in exchange for cash
b) default and the exact same process taking place, except that the banks get less for their money
in the same way that in an upturn the reverse is true (leveraging is accompanied by increasing asset prices and low default rates).
I'm not sure of the real world relevance of this primordial example but it does perhaps emphasize that giving banks cash in exchange for assets will not solve the problem, because the key is the cash shortage in the real economy and not within the banks.