What makes it inherently unstable is the relationship between liabilities (which are deposits due on demand) supported by assets of a longer maturity. My reforms would require/encourage a closer matching. Then, and this is the important part, when such enterprises fail, neither depositor nor the bank is rescued. Failures will be less frequent due to the injection of reality into the banking equation.
This problem was solved 40 or 50 years ago, at Bell Labs. Telephone companies have the same problem: how to accommodate customers who come and go at whim, using relatively fixed capital assets. I do not understand the math, but it relies on the so-called "law of large numbers" to predict that this-sized central office will support that-sized town. There is a confidence interval here... there will be outlier situations (e.g. cell phone usage after the WTC attack, Mother's Day) where some will people will not get dial tone. There is also a fair amount of waste as $40 million assets run at idle speeds from midnight to dawn.
Still, this system works remarkably well for everything from telephone service to airline scheduling, and it is used throughout our economy to find a reasonable level of assets to deploy to accommodate volatile demand. Doing the same thing with "money" makes perfect sense.
Usually the people who use this system use an alternate technology to handle the outlier situations, because just throwing more of the same sort of capital at the problem becomes extraordinarily expensive as we approach the worst-case scenario. AT&T has some "mission control" somewhere that can change call routing on the fly to deal with regional bottlenecks that pop up; the electric utilities have a grid that can do a similar thing. The regional Federal Reserve banks do that with money.
My point is that this is not some weird construction we only see in banking; it's the general solution to the problem of accommodating volatile demand with capital-intensive technologies.
It does fail at the margins; there are delays in making calls on Mother's Day, and lots of people in New York could not get dial tone on 9/11. Building everything to accommodate worst cases, however, is hugely wasteful, and in some cases can become infinitely wasteful.
There's probably a "fanatic's role in history" in this for some guy who wants to argue in favor of spending more for greater safety, but the empirical evidence says that the public does not really want to pay for perfect safety.