When banks start failing left and right, people start getting scared, taking their money out of banks, causing banks to become insolvent, causing more banks to fail, etc. When a bank fails, you lose money (absent depositor's insurance), potentially lots of it. This is not a good thing. It caused a lot of pain in 1929/1930s in the U.S., and led to the creation of the FDIC. Even 80 years later, we sort of have a cultural memory of bank failures as being a large part of causing the Great Depression in the U.S.
The FDIC is backed by the full faith and credit of the United States and just like no politician will vote to reinstitute the draft, no politician will do anything but prioritize those payments very, very high.
Remember, nobody is too big or has been around too long to be allowed to fail. Assyria fell, so can Goldman Sax.