Suppose one big bank makes a mistake, and that big bank is 50% of the market? You have a problem.
Suppose lots of little banks make the same mistake, and those little banks are 50% of the market? You have a problem. The same problem.
If banks' mistakes were independent, having lots of little banks is obviously safer. In the limit, with a very large number of very small banks, you would know exactly the same percentage of banks that would fail each and every year, so there would be no aggregate risk.
But banks' mistakes were not independent in the present crisis. A lot of banks made the same mistake at the same time.
There seem to be two types of risk: individual-specific risk, where individuals have an independent probability of making a mistake at any time; and population-specific risk, where a given percentage of all individuals has a probability of making the same mistake at the same time. Diversification into lots of little banks can protect against individual-specific risk; but cannot protect against population-specific risk.
"Put all your eggs in one basket, and watch that basket closely" would be the optimal strategy to protect against population-specific risk, since it's easier to watch one basket than many. There's probably an optimal size and number of banks, neither very large nor very small, with both types of risk.
I wonder if that optimal number is, say, somewhere around 6?
Now, how would we model those two types of risk, mathematically?