The purpose of QE1 and QE2 was to bailout the banking system and thereby, according to Bernanke, save the world economy from collapse. You can read his statements here:
Of course, his statements also mention pushing interest rates down to get credit flowing again and stimulate spending and production. But as you point out, the Fed’s actions are not consistent with that narrative. But they are consistent with a bailout of banks. In addition to the two QEs that took bad assets off the books of banks and replaced them with cash reserves, the Fed helped the banks by paying interest on reserves so that banks would not be sitting on idle cash reserves, but earning revenue to bolster their financial situation.
With QE3, there was no more talk about a banking collapse, but the policy has had similar effects as the earlier QEs. The Fed pledged to buy an additional $40 billion of Mortgage Back Securities and $45 billion in Treasuries each month. So, just like the first two QEs, QE3 has bolstered the balance sheet of the banking system without loosening credit significantly. Here is the story: