You are right if you mean that the FDIC has averted local bank runs by guaranteeing deposits. However, have you considered the other consequences?
Guarantees, especially ones obtained politically, allow the people enjoying them to take on greater amounts of risk. Banks need not match their assets and liabilities since they believe that they will not be held liable for failing to meet their obligations. As a result, banks have perpetually lent short term and borrowed long term. This has affected the structure and timing of capital investment and has enlarged the boom-bust business cycle. With the 1999 repeal of Glass-Steagall prohibitions (also enacted in 1933) on combining deposit and investment banking, this distortion to the capital structure was enlarged further. Armed with funds from deposits, investment banks were able to take on even greater degrees of leverage and move the economy that much more.
In general, political guarantees are dangerous. If the people receiving the guarantees could be regulated honestly and fairly, you might argue that they could be managed [1]. However, people receiving such guarantees tend to have resources at their disposal that allow them to effectively lobby the regulatory agencies and the ruling political bodies that govern them. The tendency, therefore, will be to reduce regulation or change its focus while retaining the guarantees.
Focusing again more specifically on the FDIC, it has succeeded in averting local bank runs. It has not averted system-wide instability. To the contrary, it has contributed to it. Faced with the danger of runs, a bank would have greater incentive to guard the value of the currency that it issues. Freed from this proximate danger, banks are able to expand the monetary base with less regard for the true fundamental wealth represented by the money. While depositors have been freed from the danger of losing their deposits, they have been subjected to greater swings in the business cycle. If allowed to go on, the day could come (some think soon, though many of these have thought so since at least the 1970’s) when there will be a run on the entire currency. People could discover that they are not certain what value is represented by a dollar and abandon it for other medium of exchange or even barter. Needless to say, this outcome would be chaotic and hardly worth the short-term gains of avoiding local runs.
In conclusion, I have laid out less visible impacts of the FDIC. I doubt that I have provided enough support to get you to fully buy in, but I fear that writing too much more would lose the interest of most readers at a site like this. Even if you can not or will not follow all of my reasoning, I hope that you will consider the complexity of society. The impacts of institutions, especially ones held up by political will, are often difficult to trace. We should be very cautious before declaring a program such as the FDIC a success. In my view, while it has achieved its stated aims, it has been horribly disruptive to society and could prove to be disastrous as it props up government money that has been made unstable.
An analysis of government-run health insurance, naturally, would focus on different specific features. But the basics are the same. Pay attention, especially, to the influence of special interest lobbies on regulatory agencies and political bodies. There will always be a push to influence large amounts of money that is politically directed. We are fooling ourselves if we think that political management can be moved away from these special interests. And again, there is a broader discussion on the economics of political management to which I have only alluded.