The banking system might be tested if people don't trust the Government to bail them out

Government debt includes bank credit that has been issued by member banks of the central bank system; private banks.

When the banking system issues a mortgage, or typical debt, this is a charge to the taxpayer. Bank debt is bad for taxpayers because it is they who will be the ones forced to pay when the liabilities are due, to the savers. It is reasonable for the taxpayer to object to banks issuing State liabilities in their name.

Has the taxpayer consented to deposit insurance? Should we be forced to pay? If there is no expectation that the State (taxpayer) will bail out the banking system, eventually, then no inflation would have resulted, it is the reason that bank credit remains valuable; a bailout is expected by the market.

The market is expecting a bailout of the banking system...

There would be a widespread bank run of the banking system, if the Government is not trusted to secure the deposits. Trust in the Government explains the absence of a (significant) bank run so far. There won't be a bank run if people trust the Government to bail them out. Lack of trust might result in a (bank) run. A lack of confidence in the Government might result in a collapse of the banking system.

A lack of faith in the Government to secure the banking system might result in the integrity of the system being tested.