Why were there numerous banks run right after the stock market crash of 1929? Why would a bank fail because the stock market crashes?
The bank runs were not caused by the crash; they were both caused by the same phenomena. In the run-up to the crash banks around the United States sent requests to city banks and the reserve banks for gold. In part due to the poor information collection system at the time, there was no way for the Fed to distinguish between a heating up economy and a true shortage of gold.
The Fed interpreted the requests for gold as a heating up economy and so contracted the money supply by 3%. This exacerbated the shortage of gold around the United States. It quickly propagated through the world financial system because of the shared gold standard. The market crashed because banks were issuing margin loans at up to 90% of value. The older system in use, before the Fed, was to create a banking suspension where banks would collect gold, but not disburse gold. The Fed made that functionally impossible forcing runs.
The thing to remember about stocks is that they are purchased with your spare money. Given a choice of eating food or buying shares in Apple, people will choose food. When the money supply contracted, funds didn't exist for businesses to continue operations, to pay employees, so people sold stocks. The sales forced margin calls, which triggered more sales, which triggered more margin calls and so forth.
The NY Fed did provide liquidity at the beginning of the crisis, but the Board of Governors were still convinced that the contraction was necessary and forced the end of the NY Fed's liquidity provision.
The stock market was the thermometer, not the cause. The system had become sick.
To be fair to the Fed, it was itself part of a much larger set of global events without an information base to act on. It was also true for the Bank of England or the main European banks. The gold standard propagated economic issues in ways that were not understood at the time.
When World War II ended the nations of the world created a variety of treaty organizations such as the United Nations and the OECD to prevent such a sequence of events happening again. The problem is that restricting those events from happening means that some large business owners miss out on opportunities and so there is pressure to dissolve or weaken the treaty organizations. In addition, the major nations sometimes endanger the system. You can see this very directly through the actions of the Trump administration.
The postwar era institutions really do need reform as the threats have changed, but the political will to protect the world doesn't play well in local political processes.
A bank has \$1,000,000 in deposits but has lent out \$900,000 of that, so it only holds \$100,000 cash. So long as a lot of people don't need money at once there's no problem, but how can you, as a depositor, be confident that the bank will have money when you need it? Better to go and withdraw it all right now, just to be safe. Now get about half the bank's customers thinking this way and there is a run on the bank.
This scenario, more or less, played out dozens of times during the '29 stock market crash, and it is this that made the otherwise esoteric stock market relevant to ordinary citizens (and severely affected their economic circumstances).
It was a much different time back then. Gold and silver were considered money. A majority of banks where not members of the Federal Reserve. The business of creating money by banks was to count deposits of other banks' notes as cash. So during the roaring twenties there was a huge creation of
money. A demand on deposit for gold and silver was the norm. The stock market bubble was fueled by this money creation. Everyone was buying the market on credit. The market failed and that debt could not be paid back with money that did not exist. As a majority of deposits were counted as notes rather than cash, banks had no way to meet the demands for deposits.
These days, because of the Glass-Steagall act, commercial and investment banking are separate entities. You would keep your deposits at a commercial bank where risky investments of your deposit are not allowed. Also, these days, there is no demand on deposits in gold and silver; only Federal Reserve Notes.
The "Roaring 20s" was a period of overheating (in today's lingo). Very similar to the 'dot-com' bubble in the 1990s. So the stock market was a symptom, not a cause. In both cases the bubble burst (1929 and 2000 respectively). But in the dot-com case, the Fed lowered interest rates and increased liquidity, thus avoiding a depression.
It's amazing to think that the entire Great Depression could have been avoided easily. All the Fed had to do was print up a bunch of money and lend it to the banks. Then the banks could have given cash to all depositors who were withdrawing. Then a few months later, when people are no longer freaking out, they would put their money back into the banks, the banks repay the Fed, and you're back to normal. (Or maybe an ordinary recession, like post-2000).