Before the internet era, it was much more difficult to buy funds, stocks etc. Especially when you go as far back as to the crash of 1929.
So when people stored all their money in bank accounts, it meant a tragedy when the bank went bankrupt and the money was lost (See: the
Great Depression). So state-funded deposit insurances made very much sense.
Nowadays it is really easy to buy funds and stocks, so there is no reason to keep all your fortunes in a bank account. The question is then: Why should tax-payers cover for the banks gamblings?
Instead of these insurances one should have warning labels, similar to warning labels on cigarette packages (
Tobacco_packaging_warning_messages) That is a certain portion, like a third, of the front and backside area of the package is up to the authority to decide how it should look. And they have labels like "Smoking kills" etc.
Imagine all bank account statements, papers from the bank etc, each sheet having a big warning label "If the bank goes bankrupt, these money will be lost" or something like that.