distinction between a bank and an investment bank
Here in the US "banks" were, up to a time, required to keep their depository institution functions separate from their investment banking functions. The depository stuff is pretty low risk. The investment stuff is pretty high risk. We had to bail these banks out for the reasons you cite because the two got mixed: they were doing investment banking with depositors' funds. If they'd stayed separate we'd have been able to allow the investment banking to fail, as it should have done given the stupid risks they were taking, while the depositors remained in the clear.