Trickle Down
In a discussion with a friend, we came up with another reason why letting banks fail would have been a good idea—it would have helped borrowers who had overextended themselves.
When a lender fails, another institution generally acquires its assets (including mortgages) at fire-sale prices. If such an institution acquires a sub-prime loan at pennies on the dollar, it is generally willing to negotiate the terms of that loan, in order to reduce the risk of it defaulting. What this means is that a borrower who cannot make payments can get some leeway on the interest rate, payment frequency, and possibly even the principal.
Turns out this is exactly what Bank of America is doing with sub-prime loans it acquired with Countrywide financial.