Can a bank sue someone that starts a bank run that destroys the bank?
No (assuming, of course, as is the usual case, that the person who starts the bank run is not engaged in perpetrating a defamatory falsehood).
Most bank runs are, and certainly the Silicon Valley Bank bank run was, based upon wide disclosure of a true fact. In the case of SVB, the bank run was triggered by the fact that its balance sheets failed to reflect the true value of fixed nominal rate bonds that it held as assets.
In the usual case, a lawsuit also isn't a very helpful option to a bank that suffers a bank run. In the case of SVB, the bank had a book value (which is often a fair measure of a bank's value since its assets are so monetized) of $34 billion which was reduced to a pittance by the run on it. Even if someone who started a run on the bank had a moderately high net worth of $3.4 million that could be collected in a money judgment, that would cover a mere 0.01% of the loss to the bank, and there would be serious issues over the causation of any loss (i.e. how much of the losses suffered bound to occur sooner or later anyway due to causes unrelated to someone who triggered a panic).
Another fine point of procedure is that when a bank becomes insolvent, it is promptly taken over by the FDIC or similar regulatory agency, which installs a receiver. This makes it effectively impossible for the bank itself to sue anyone. If the bank would otherwise have had a right to sue, the receiver for the bank would have the right to sue rather than the bank itself. But, this subtly while not irrelevant, doesn't capture the core reason for the question.