I don't agree. Bank runs caused by liquidity issues are a solved problem. Bank runs caused by insufficient equity (say due to bad loans) cannot be solved.
With Bitcoin, a bank run would look like an attempt to withdraw larger amounts of Bitcoins quickly, rather than queues forming in front of the banks. If there was a problem with withdrawals, people would start complaining in the forum, the news would pick it up and it could escalate very quickly. Even now when the amounts are, on global scale, negligible, it does not even take 24 hours for the word to spread.
The issue of liquidity is not solved at all. FRB works because banks issue short-maturity (even zero maturity) instruments, but the maturity of the loans they issue is higher (can be years). They carry the risk for this difference. If they were forced to liquidate the loans prematurely, they would need to take a cut, and this would result in undercapitalisation. Furthermore, as Taleb for example convincingly argues in The Black Swan, a lot of risk is not correctly modeled.
This is a smaller problem if FRB is merely a method of bringing together investors and creditors. People normally do not expect to withdraw their deposits immediately. If however those instruments are also used as a medium of exchange, people do expect to be able to use them for payment right away, and if this does not work, it has a direct impact on their business or lifestyle, and aggravates the panic. People can't pay rent, food, their suppliers or employees. That's an immediate problem.
I would say that most people depositing into traditional banks do not think of themselves as investors, or at least they don't think of their checking account as an investment.
There is a duration mismatch problem that is fundamental. If a bank loans out money for a 30 year mortgage, but people want their money back sooner, the bank is screwed. The bank is "good for it", but can't pay out today. The modern solution is for a central bank to be able to create money out of thin air and lend it to the bank.
Since a bitcoin bank can't create money out of thin air, it always faces the possibility of a run. Bitcoin banks could band together with an agreement to loan each other bitcoins as needed, and this would mitigate the problem during isolated or limited bank runs, but if everyone everywhere gets spooked at the same time, the game is up.
The solution is to be clear on both sides about repayment timelines. The bank could not use demand deposits for duration loans, and the customers could not ask for early withdrawals on duration deposits. Clever banks would attempt to mitigate this by creating markets for duration deposits, giving people an option to cash out early by selling the certificate to someone else at a discount (penalty), but it would need to be clearly understood on both sides that this system is not a promise of liquidity, but a last resort.