"Arguably, the most significant and plausible financial
stability risk of a general purpose CBDC is that
it can facilitate a flight away from private financial
institutions and markets towards the central bank. Faced
with systemic financial stress, households and other agents
in both advanced and emerging market
economies tend to suddenly shift their deposits towards
financial institutions perceived to be safer and/or
into government securities. Of course, agents could always
flee towards the central bank by holding more
cash. But a CBDC could allow for “digital runs” towards the
central bank with unprecedented speed and
scale. Even in the presence of deposit insurance, the
stability of retail funding could weaken because a
risk-free CBDC provides a very safe alternative."
The Bank for International Settlements (BIS)  is not the "world's central bank." It isn't even a central bank. It has no monopoly on any monetary base and does not act as a lender of last resort .
The reason why Bitcoin will cause the mother of all bank runs is because Bitcoin will eat up banks' liquidity that is desperately needed to keep the leverage ratios in check. Rather simply, because banks take in deposits and then generate revenue, by extracting the value of your capital, by lending it out to other people at interest. This is the most common form of money creation in the economy (with other people focusing on T Bills, quantitative easing, etc)
If people continue to dump dollars/fiat (which are only liquid if your bank doesn't screw up and over leverage themselves) for money that is always liquid (Bitcoin) then essentially the liquidity that banks rely on dries up, their leveraging goes up, and the slightest market hiccup causes an intra/inter-bank liquidity-failure cascade. 
Given that typical leverage ratios are ~3-5% (or banks only have 1 dollar for ever 33 dollars they say they have) and that the total supply of M1 is ~4 Trillion , the balance they have available (5%*2e12=$200 billion) is hitting the first threshold of concern because Bitcoin's market cap is ~$150 billion and cryptocurrencies in total are $360 Billion (though volume would be a better indication of flows). No wonder central banks are literally conspiring to smear Bitcoin rather than admit their own incompetence. Once the Bitcoin/cryptocurrency market caps approach ~$800 billion (equivalent liquidity of M2 leverage) the sparks are really going to fly in liquidity problems with banks.
Will banks avoid this? My guess is no, because such a CBDC won't allow the squish that banks need to cook books or mask liquidity problems as "Crises of confidence" and the competitive forces demand higher and higher amounts of leverage and that the government has colluded with the banking industry to constantly socialize their losses, while the banking sector privatizes gains. For more reading check out, The End of Alchemy 
Which do you think is more likely Bitcoin is going to stop working or banks will under-engineer their financial safety factors and fail (like they've habitually done)? Hopefully you have your skin in the game your gut tells you is going to happen.
Why would anyone choose to hold a coin controlled by the central banks if it’s the system created by the central banks they’re trying to escape? The safest and most proven cryptocurrency is Bitcoin - and that’s where the flight to safety would be.
In two comments here you've called USD unstable and Bitcoin stable. Can you explain what stability of a currency means to you? Because it appears to be very different from how the rest of us define it.
Anyone caught in a bank run only has themselves to blame. It's their responsibility to not deposit their money in an over-leveraged bank. Unfortunately due to FDIC insurance, consumers now spend more time researching the quality of the LCD monitors on the market, when deciding which one to buy, than the future financial solvency of banks, when deciding which one to trust their money to.
It is not an unfortunate fact -- civilization moves forward when we increase the number of things we can do without thinking.
FDIC, as the underwriter, has the power not to insure a manifestly poorly-run bank. Thus, the FDIC mark is a defacto sign that a bank can be trusted by a consumer.
I would much rather have FDIC (or a system of robust private bank insurance) than need to vet each prospective bank's balance sheet, employees, and loans when choosing to open a personal checking account.
If you draw all of the system out you'll see it's no longer tangible and that's why FDIC allows for 99 years to make good on that promise. I think the math works out that inflation alone in a system that survives 99 years can recoup any system that collects an average of 30% recirculation
Being responsible for yourself doesn't mean you don't delegate. It means you have to stay on top of who your delegate is. FDIC means you simply tax others when you screw up. The insurance premium has no relationship with your risk profile.
Responding to below:
>>Successfully rescuing a bank from a run is not only a benefit to society as a whole but often makes a small net profit.
This is the free lunch fallacy used to justify involuntary income redistribution. If it were indeed profitable on average, the market would do the bailing out.
Bailing out the irresponsible prevents the evolutionary process of weeding out reckless and incompetent business models and behavioural patterns from working, while unfairly burdening those who are competent in their management of capital, which inhibits the expansion of their influence.