The FDIC has already come perilously close to running out of money once. It can happen again.
I believe risk can be moved around, but it can't be eliminated. This is probably fairly mainstream Wall Street analysis, too. I also believe that all attempts to move around risk must also increase it on the net, e.g., if you do the extremely popular financial move of taking a modest chance of modest loss and stuffing all the risk into a small chance of total loss and then hoping that just never happens, that the total risk has increased. This is probably less mathematically provable, and depends on a lot of assumptions about how one compares various classes of risks. (I base my thinking on a variant of what you might call the "more or less efficient market hypothesis"; if there was a way to net reduce risk overall, it would probably already have been taken.)
The bank abstraction you refer to is in my opinion in that class of "extremely popular financial moves". Yes, it removes the modest risk of partial loss, but it moves the risk entirely into a small chance of total loss when the entire system comes down instead. I worry about the net effect of millions of little financial transaction that all perform that particular risk management move. It ends up creating a lot of easily-hidden correlations in the system, and no one entity (let alone person) can have a view of the whole situation.
In the end, the risk is moved around, but the bank run is still a possibility and no amount of financial abstraction can truly remove the fact that the liquidity term mismatch is a fundamental aspect of such an operation that can't be erased, only managed.