The banks absolutely did give mortgages to unqualified borrowers. In fact, almost ALL the mortgages they originated and foisted off on the GSEs were sub-standard. That's why the DOJ went after them and extracted huge settlements.
And your assumption regarding why the government told the GSEs to purchase lousy paper had nothing to do with saving the banks. It had to do with saving the housing market because banks had no liquidity for new mortgages until they cleansed their books of the old loans. Banks are required by FDIC rules to maintain a liquidity balance between loans and deposits.
The banks were "righted" by TARP actions, not saved by the GSEs. That's history. Read up, if you wish to understand the truth.
JMHO.
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