Banks can't fail because of deposit insurance

The Government are employees of the central bank.

Because people trust deposit insurance, from the Government they will leave their money in the bank. People 'trust' the banks for no other reason than they have deposit insurance, there is no other reason that they would use an insolvent bank. People don't value the credit of insolvent firms unless they have a banking licence. The banks are trusted because of deposit insurance, they (the banks) cannot go out of business.