Remember that, during the financial crisis, governments had to save many specialised banks in order to  BIS central bankers’ speeches  1  prevent systemic disruptions. The Lehman bankruptcy caused precisely such contagion, but not because customer deposits were at risk. Rather, it had many connections to other financial institutions including, above all, many counterparties in derivative transactions.
Essentially the same applied to AIG which was not even a bank at all, but an insurer. Even if there were a clean organisational separation, the interconnectedness of the financial sector and the resulting systemic importance of individual institutions would not be entirely eliminated. Economic linkages would remain.