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.
Furthermore I'd argue that insuring deposit banks through the asset-limited FDIC is yet another setup to socialize catastrophic losses. If bank deposits were guaranteed by the Fed directly, the Fed could print an unlimited number of dollars to make depositors whole, thus completely eliminating the risk that a USD-denominated deposit simply vanished. (The realized damage would instead be expressed as monetary inflation)
The FDIC is backed by the full faith and credit of the United States. It legally cannot run out of money.
Its fees are mostly an accountability mechanism on the bureaucracy. If the FDIC burns through its reserves and has to ask the Treasury for money, it failed at risk management. Some folks may get fired. But the bills will be paid.
The FDIC is legally backed by the full faith and credit of the United States, its liabilities beyond its own resources, to the extent that such may exist, are debts of the United States.
Which has a fundamental power to mint US currency, and thus cannot meaningfully “run out” of it.
On the other hand, in order to do that, you have to pay them a higher interest rate to make it worthwhile. And rates have been driven to the floor.
Ponzi schemes are an extreme example, because the fund is illiquid because it lied about its investments and simply stole the funds, meaning there's absolutely no way to cover.
But there's a lot of ways this can occur. This basically describes a "bank run"; you nominally have an agreement you can withdraw your cash at any time, but if everybody does it at once there's a problem because the bank is using it to do things like fund mortgages, which can not simply be called back in instantly if needed. Even if the bank has the right to do that, and for various things it sometimes does (taking "financial instrument" generally and not just "US mortgage"), it would still be squeezing blood from a stone; they can demand it but it doesn't mean they'll get it.
In recent news, Blackrock suspended withdrawing from a UK real estate fund, because investors have been withdrawing at a rate that would require them to liquidate their holdings at fire sale rates, further depressing the fund's value: https://uk.finance.yahoo.com/news/blackrock-halts-withdrawal... Same sort of thing. No "Ponzi" scheme here; there's some legitimate financial stress, but that stress would only be exacerbated by letting everyone withdraw. (Whether you agree that it is justified to halt withdrawals in this case or not, I'm just using it as an example of the generality of this structure.)
I can't pull it up quickly in a news article because I'm not coming up with the right search terms to pull it out of the noise of constant financial news, but when Janet Yellen was Treasury Secretary, she had floated a trial balloon about trying to fix this incentive problem with bank runs against real banks with the same sort of clawback scheme, the idea being to disincentivize a bank run in the first place by making it so you don't get the pattern where the first few people get all their stuff and everybody else loses everything, which is a huge contributor to the run occurring in the first place. The game theory on this one gets a bit complicated if you think about it. e.g., OK, if everybody else is going to sit tight because this scheme incentivizes them to stay in the bank, then it's safe for me to withdraw everything because the bank run was prevented in the first place. If they then hold things together "long enough", then I can say I did it for my own reasons, not the bank run that was delayed for six months while everyone else sat tight, before an ultimate collapse. But if everybody thinks that way... etc. Not clear to me whether this can actually prevent bank runs, which I mean straight, not as a weak sarcastic "no this obviously wouldn't work". Not clear. Complicated analysis.