I would just add that it also did not prevent runs on depository institutions throughout the volatile 19th century. During that century, we had a system of unit banks since branching was restricted for all federally chartered banks and most state-chartered banks. In fact, by 1920, we had some 30,000 banks operating in the country. By 1920, the state of Nebraska had a bank for every 1,000 people! What did all this emphasis on small banks accomplish? Not much in terms of stability: The U.S. suffered major banking panics regularly throughout the 19th century and into the early 20th century (8 systemic crises between the 1830s and the creation of the Federal Reserve). What about our Canadian neighbors? Zero.
So what's the difference? Why did the U.S. have such regular upheaval while our northern neighbors avoided it? Renee Halton does a good job of describing the differences in this paper, but the main idea can be found in O.M.W. Sprague's classic analysis of American banking (done on behalf of the National Monetary Commission). Sprague argued that the unit banking structure was the main flaw in the United States; Canada had no such restrictions on branching.
While there were obviously many factors other than branching at play in America's long history with financial instability, we should be careful about longing for a return to the small unit banks of the past.