This issue resides in the sphere on my professional knowledge.
To answer your question on payouts: from 1934-2002 (a date cutoff picked to try to stay within the 20 year rule, if necessary), the FDIC data shows that the FDIC, it's defunct sister program the FSLIC (Federal Savings and Loan Insurance Corporation), and the Resolution Trust Corporation (created to clean up the mess that was the Savings and Loan Crisis after the FSLIC had become insolvent and incapable of resolving the problem) have spent a total of $106 billion resolving failed banks. Including the 20 years since would increase that total by roughly 70%. This money is spent to make depositors whole, either directly (by paying out depositors) or indirectly (eg by offering loss share agreements to third parties to get a better up front price on the failed bank's assets).
The FDIC will seize a bank when it determines that the value of its assets is no longer sufficient to meet its debt obligations. There is some judgement involved here as some banks that remain open during a period of duress can appear to be in worse shape than other banks that the FDIC closes.Such decisions are made by the FDIC examiners based on their detailed knowledge of the bank and its financial and management situation. In general, the FDIC attempts to manage the shut down so that all creditors, other than equity, are made as whole as possible, however insured deposits get priority and only if there is money left over do other creditors receive payment. Equity investors in banks very rarely receive anything. It is possible for the FDIC to resolve a bank failure without losing any money, although this was achieved only 57 times in 2,965 closures from 1934-2002.
The FDIC has been competently run in its history and has never needed a bailout from the US government, unlike its abolished sister the FSLIC.
From this perspective, I would argue that the professor's statement is false. The FDIC has proven to be needed.
Regarding bank runs: It is true that bank runs are quite rare since the establishment of deposit insurance corporations in the US. It is possible this is due to the calming influence of the FDIC. The way the FDIC handles closures is also part of this picture. The true financial situation of a bank can be very opaque for the common person. Even finance professionals working within a bank can sometimes fail to realize that their employer is on the brink of failure. When the FDIC closes a bank it does so with no public notice, it takes over after close of business on a Friday and, if everything goes well,opens on Monday under new ownership. The FDIC strives to have a buyer lined up before closing the bank. "Buyer" here is doing a lot of work as they often pay no money but agree to take over the operations in exchange for taking on the toxic assets (possibly with a loss share agreement with the FDIC). So the FDIC does not leave any time for a bank run to start.The closure usually comes from out of the blue from the perspective of the branch visitor.
This does not mean there have been no bank runs in the USA. Home State Savings Bank had a real old school "get in a line" bank run in 1985. The failure of Home State was too large for the Ohio state run deposit guarantee fund and cause the governor to close all other banks covered by the state's insurance for 3 days to stave off other bank runs.
More recently (and apologies if this is a 20 year no no), 2008 saw three bank runs to my knowledge. IndyMac lost over 7% of its deposit base after Senator Schumer publicly published a letter questioning the viability of the bank. Washington Mutual became the largest bank failure by asset size in US history when it suffered an epic multi day bank run where it had over $15 billion in deposits withdrawn. The WaMu situation was serious enough that the FDIC closed the bank on a Thursday. (JP Morgan acquired WaMu for an excellent price, but the FDIC lost no money here actually). The failure of Washington Mutual may have triggered a bank run at Wachovia (which engaged in similar lending practices), but what is certain is that Wachovia saw $5B leave the bank in a short period of time. The FDIC did not technically close Wachovia but did help quickly arrange a transaction where Wachovia sold its entire operation to Wells Fargo.
So, there ARE STILL bank runs, but they are admittedly quite rare. It is impossible to prove that there would have been more bank runs without the existence of the FDIC, however the hypothesis does seem to be reasonable given that the way the FDIC manages bank closures does not leave much opportunity for such events.
edited for formatting and some grammatical issues.