This is where the "lender of last resort", i.e. the government, comes in. Depositor losses are sufficiently bad for the economy as a whole that it makes sense to prevent them at almost any cost.

The UK ended up owning RBS: https://commonslibrary.parliament.uk/royal-bank-of-scotland-...

Ireland got hit pretty hard: https://www.irishtimes.com/business/financial-services/2022/... but did at least manage to send Sean Quinn to actual jail.

Cyprus had too large and too dodgy a banking sector for its government to cover, so there depositors did incur real losses ("haircut").