How would banking in EU change if taxpayers withdrew their support for Deposit Guarantee Scheme?
Based on the discussion in comments, the core of your question appears to be "What would happen if European governments stopped providing deposit insurance?"

The question and answer about the US deposit insurance program here provides useful background, but the short answer would be that we would expect bank runs to be more likely, though how much more likely would be a function of the institutional relationships between banks, banks' access to liquidity backstops, the extent of bank regulation, and economic fundamentals. This is because (as discussed in the linked answer) private deposit insurance doesn't entirely relieve depositors of monitoring burdens, and private deposit insurers lack the ability to discipline the banks they insure without potentially creating bank runs.

Theory aside, it's not difficult to find examples (particularly historical examples) where deposit insurance has not existed, and while bank runs were generally more likely in those cases, they were not always common, as emphasized by Calomiris and Gorton (1991). Institutional arrangements such as credit mutuals, where shares in the depository institution are allocated based on deposits, can mitigate run risk, for example. A decent review of banking from an industrial organization view can be found in Neuberger (1998).