In every major economic crisis of the past century, the failure of banks to adequately police themselves by eschewing financially risky behavior has been a factor. That was certainly the case during the 1920s, a period of profligate spending and investment with insufficient regard to the considerable risks involved in the kinds of financial transactions then prevalent. One such kind of financial transaction was the widespread practice of buying stocks "on the margin." Buying stocks "on the margin" simply referred to the purchase of stocks with borrowed money rather than with money that was on hand, as in a savings account or in any other highly liquid form. When borrowed money is used to purchase something that has a value that can fluctuate wildly, then the amount of risk involved increases dramatically. During the so-called "Roaring Twenties," investments made with borrowed money came to represent the overwhelming majority of stock purchases. When stock prices began to slide, then, investors and banks alike found themselves unprotected against defaults, and those defaults were numerous.
The Great Depression had a number of causes, as is regularly pointed out, but there is no question that the widespread practice of buying "on the margin" constituted a significant factor. As noted above, banking practices involving loans with inadequate collateral and loans made for political or personal reasons can seriously weaken the banking structure of entire countries, as happened most prominently in Japan starting in 1990. The United States learned some invaluable lessons from the Great Depression, but even the value of those lessons diminished with the passage of time, as became evident during the financial crises of the late 1990s and in 2007–2008.
Buying stocks on the margin is fine in prosperous times when confidence in the economy is high. When that confidence begins to diminish, however, and when stock prices begin to decrease, the money needed by those who purchased stocks on the margin is no longer available. The value of the investment has dropped, and the debt incurred in the stock's purchase becomes untenable. It is, in a way, a vicious cycle of economic activity, and one that played an important role in causing the Great Depression that followed the stock market crash of 1929.
As you say, the stock market crash did not cause the Depression all by itself. But it did help, and the buying of stocks on margin was a major reason that it did so.
Buying of stocks on margin refers to the practice of borrowing money to buy stocks. If the stock price goes up, you're fine because you can pay back what you borrowed. If the stock price goes down, you have to pay back the debt and have no money with which to do so.
After the crash, the stock prices were way down. So now all sorts of people owed money for their margin buying. They couldn't pay the loans back. This meant banks went broke as their loans all went bad. When this happened, depositors (pre-FDIC) lost their money. Businesses couldn't borrow either. Between all of these things, the economy took a very hard hit because of the margin buying followed by the crash.