Perhaps it would have been better to let the private banks fail and let those who took the risks go down with them?

The U.S. could have established a public bank to provide loans with the bailout money instead. A "public option" to the bailout, if you will.

The new bank cost could have been less expensive than the $700 billion bailout since the "public" banks could have done what private banks do using the fractional-reserve system to "create" money out of nothing.

The $700 billion was borrowed. Can anyone explain why should a government have to borrow its own fiat money?