The problem seems to me was having banks exist that were "too big to fail", where they could make investment decisions based on the assumption the government would bail them out if they failed.

this wasn't the case. banks and mortgage companies were allowed to make terrible decisions because there was insufficient regulation and because they were greedy...not because they knew they'd be bailed out. they were only bailed out to prevent a further collapse of the financial system. even if what you say is true, and they made bad decisions based on an assumption that they'd get bailed out, those decisions were motivated by greed and a lack of consequences