All this assumes a normal economy where a bank has no incentive to hold reserves in excess of what it is legally required to do and where treasuries and cash are not good substitutes for each other. In a crisis, the economy is not in a normal situation. Interest rates could be touching zero (as they are now) and treasuries and cash become almost substitutable. If the central bank now starts buying treasuries for cash from banks, the banks will happily take the cash and sit on it. They will keep reserves much in excess of what they are required to, as they don't have confidence in private assets, treasuries are equivalent to cash and the possibility and fear of a bank run (or a loss of confidence in the banks' assets) makes it prudent to keep more reserves than is mandatory.
The part where banks consider the excess cash as a substitute for treasuries and hold on to it (just like they were holding on to treasuries) is the liquidity trap. The part where banks hold excess reserves due to fear, let's call that a 'solvency trap'.
To get past this situation, Keynesians recommend fiscal stimulus. The government issues new treasuries, this increases the interest rate on treasuries, people buy treasuries (through banks) and the government has cash to burn. Of course, some of the excess cash will also be used up to buy existing treasuries from the central bank itself, i.e fiscal stimulus will also have the effect that a mild monetary contraction in the form of an interest rate hike does. This can be avoided by concurrent monetary expansion by the central bank.
Irrespective of any such concurrent monetary policy measure, fiscal expansion ensures one critical thing at least - the government borrows from lenders because the private sector is not able to. The average willingness to spend cash has increased. The government can then spend the money in various ways. It can choose to invest in infrastructure and aim at the positive network effects to boost the economy. It can reduce the debt burden of households and businesses by re-capitalizing them. Or, it can decide that it has no competence in choosing and privileging specific households and businesses and instead just aim at getting the stimulus to work through the banking system by giving cash back to the banks.
But lack of cash with banks was never the problem - the willingness to get it moving was! And the government is not going to just make a donation to the banks. So what happens in a fiscal stimulus that operates through banks anyway?
Well this is what I think happens. Government issues new debt, lenders of all hues move cash from banks to treasuries. The balance sheet of the banking system shrinks. The banks also buy up some more treasuries from the central bank. Now the government is going to give this cash back to the banks in exchange for something. What is this something?
It won't be treasuries, that will just be a monetary expansion and fiscal contraction. It won't be true short term debt - that's just cash back to the bank and all that has happened is that the bank owes money to the government instead of investors and the government in turn owes money to the investors. It can, however, be long-term debt, or short term debt with the surety that it will be rolled over (which is just cheap long-term debt) or equity. If it is long term debt, the asset liability mismatch of the bank is corrected to an extent. If it is equity, the bank is unlevered a little without a shrinking of the balance sheet. In either case, fiscal stimulus that operates through the banking system fundamentally improves the capital structure and hence the solvency of the banking system. This is bound to reduce the tendency to hold excess reserves and get the money circulating faster.
Conventional monetary policy then just involves a shift between two safe assets of the banking system - it cannot be expected to work when there is substitutability between these assets. Unconventional monetary policy, in which the central bank buys private assets from banks in exchange for cash, also exchanges one asset for another but these are not substitutable. It can, in principle, work.
Bank-driven fiscal policy effectively involves tinkering with the liability structure of the banking system. It can be a way out of the liquidity trap as well as the solvency trap. In fact, it can help the economy escape out of the liquidity trap precisely by getting out of the solvency trap.