How would banking in EU change if taxpayers withdrew their support for Deposit Guarantee Scheme? Disclaimer

The question pertains mostly to EU but I believe counterparts to Deposits Guarantee Scheme exist world-wide.

Assumption

Let’s suppose that support for liberal ideas and free market rises in European nations what results in taxpayers withdrawing their support for majority of state’s interventions in the free market, including Deposit Guarantee Schemes. Taxpayers would also demand that the minimum capital to open one’s own bank would be dropped from 5mln Euro to 100k Euro, so that the competition in the banking sector is at a much higher level that it is nowadays which is an interesting topic in itself but let’s just focus on DGS exclusively.
I am far from judging who would benefit on that or whether such a situation is even likely - it’s simply an axiom for this question.

Question

Now, given that Fractional Reserve Banking model makes it impossible for banks to guarantee even a small fraction of deposits in their own capital as clearly banks juggle money supplies rows of magnitude higher than they own, what can we reasonable expect to happen if taxpayers turn their back on banks and demand banks to provide their own guarantees?

I presume that once the taxpayers’ guarantees are gone, for banks to be perceived as a safe place, they would have to own capital that at least equals the deposits to be handed over to them?

Otherwise, I cannot see what would people base their trust to banks on?

References

Directive 94/19/EC on DGS

DGS FAQ
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).