I don't think that's the case. The people who hold more than £100,000 with Northern Rock don't hold them as transactions balances. For many it represents their life's savings. I got one anxious e-mail from someone whose mother was in a nursing home, had all her savings in an account with Northern Rock (as deposits of one kind or another) and paid her nursing home premia from that account. Those were not transactions balances that would be held in a narrow bank.
Would moving savings that are not true transactions balances out of deposit accounts - this is what would happen if narrow banking were introduced - and having narrow banks free of run risk eliminate or weaken the ability of savers (the (former) depositors) to extract a free guarantee of their savings from the state?
Small savers, especially those saving for retirement or already retired and living off retirement savings, want their savings to be safe. Telling them that the world is an unsafe place cuts no ice. They also want a 'decent' return on their savings. It so happens that the decent safe rate of return they aspire to exceeds the risk-free rate of interest the economy is generating. The small savers are therefore looking for a handout through the state from their fellow tax payers. If the introduction of narrow banking were to cause these small savers to invest their retirement savings in unit trusts or other non-deposit investment vehicles, the political pressure to get these investments guaranteed by the government, if there were a threat to the value of these investments, would be comparable to what we see today with the deposits.
You would not see a run on the unit trust headquarters, but you would see demonstrations of grey and blue-haired pensioners outside Parliament and petitions at 10 Downing Street.
Is there something uniquely intimidating to politicians about a long line outside a bank of depositors desperate to take their money out? It is interesting to speculate why this would be. The continued withdrawal of Northern Rock's deposits, once the Liquidity Support Facility ('credit line') was in place, no longer had any impact on Northern Rock's ability to continue functioning. By drawing on the credit line (allegedly uncapped and open-ended!), it could do without depositors completely. Using the credit line would (I hope) be more expensive than raising funds by retaining deposits or attracting new ones, but that only impacts on Northern Rock's shareholders. It is hard to believe that the deposit guarantee was provided to support Northern Rock shareholders.
What about fears of contagion to other UK banks? With the Liquidity Support Facility extended to these other banks and building societies also (as they all are no doubt solvent), they too could continue to function without depositors and deposits. There would be no threat to the stability of the UK banking system, even though there were lines around the block outside each branch office of every UK bank and building society.
We would have achieved half of the move towards narrow banking through this non-systemically dangerous general bank run. Clearly, deposits would no longer be available for transactions purposes, but this would be a nuisance, not a disaster. Cash, travellers cheques, transferable negotiable bills of exchange and other similar instruments would soon take over the role of transactions medium from the defunct deposits. If narrow banking were nevertheless deemed desirable, we could move to the narrowest form of narrow banking by giving every UK household and business a non-interest-bearing account with the Bank of England, an account that could be accessed through, say, any post office or sub-post office in the land. That's the retail payments system taken care of.
So if there is an effective Lender of Last Resort, deposit insurance is redundant from the point of view of banking sector survival and financial stability. Deposit insurance is also not sufficient to allow a solvent but illiquid institution like Northern Rock to survive, as it was Northern Rock's inability to roll over its maturing non-deposit liabilities that was causing it trouble.
If the deposit insurance were to extend to new accounts as well - it does not, of course, in the case of the Liquidity Support Facility, although the LSF covers new deposits in existing accounts, as well as a whole list of other unsecured creditors who don't hold retail deposits - the wholesale market funding-challenged bank could offer such outrageously high interest rates on its deposits, that it might be able to fund itself entirely through deposits! Indeed, that option is still open to other banks that have not yet sought the shelter of the LSF, but know that the LSF will be available to them should they get into trouble in the future. Any bank experiencing trouble funding itself in the wholesale markets, other than Northern Rock, could simply offer wildly excessive interest rates on its deposits to buy itself more time. Depositors know that there will be ex-post deposit insurance should the bank not be able to service the deposits out of its own resources. A great incentive system has been created.
In summary: if there is a LOLR, deposit insurance is neither necessary nor sufficient for banking and financial stability. Unless the sight of long lines outside the banks would have significant negative effects on consumer and/or business confidence, there are no macroeconomic stability arguments for deposit insurance provided there is an LOLR. Why there would be significant adverse effects on confidence from a bank run that would not threaten the survival of the bank or financial stability is not clear. The British like to queue.