The liquidity being provided by the central banks is of the very short term kind (ie overnight) - it's just to help banks manage their till, if you will. It does nothing to solve the state of their balance sheets, it simply prevents them from collapsing overnight because other banks don't want to entrust one another with their cash.

The way things are cannot last for ever, but it can last for a while, as banks fund themselves mostly ia the Centrla Bank instead of via the interbank market.

This Forbes article gives an idea of the size of the respective markets:

If all the recent hysterical chatter about lending being "frozen" or "shut down" refers to anything real, it is not about banks loans (through Sept. 17) but about such arcane financial markets as asset-backed commercial paper or loans between banks. But this too is mainly about financial firms, not Main Street. Non-financial commercial paper increased from $156 billion at the start of the year to more than $204 billion from Sept. 3 to Sept. 17, dipping only modestly since then.

Economic journalists seem oddly fascinated with the last column of the table--interbank loans from one to another (aside from fed funds). "Banks won't even lend to each other," said a TV reporter, "so how can we expect them to loan to business or consumers?" But interbank loans are obviously tiny, and banks rightly regard lending to other banks more risky than lending to Main Street. There is no reason to expect the minuscule flow of interbank loans to determine consumer and business loans. That little tail can't wag the big dog.

That paragraph is slightly misleading, because the interbank market may be small, but it plays a vital role in ensuring that the whole things functions - exactly like the till in a store: the cash in the tills is only a small part of assets, but imagine what would happen to business if the cashier had no change whatsoever, or refused to give back what it had.

In the long run, we're all dead. John Maynard Keynes