Sure, but allowing half the banks to fail would not have lead to a little dip. The collapse of Lehmans was very damaging for many perfectly healthy, well managed firms that depended on them. The only reason the effects were not worse was that there were other banks around to pick up the pieces, but if the domino effect had been allowed to play itself out the dramatic loss of capacity in the banking sector would have been truly catastrophic.
Note that the bailout money wasn't simply gifted with no strings, it was leant and paid back with a profit to the tax payer. Of course QE had it's costs and they were very high, there were other costs as well, but they're often exaggerated.
Paying off everyone's mortgages sounds lovely, how nice. In reality everyone would have rushed out and re-mortgaged their properties again, attempted to re-invest or spend their wind fall leading to massive house price inflation and a titanic credit binge the likes of which the world has never seen. The consequences would have been apocalyptic.