I recently watched a documentary called 'The Big Panic' about Paulson, Geitner, Bernanke etc. and how they handled the 2008 crash. They might've not done everything right but at that time they made the right call to 'save the system' and tried to limit the damage as all three had studied economic crises, recessions and whatnot. Bush was also interviewed and he didn't like the bailouts but he said that it was more important than his own popularity or any election and decided to do whatever he could as President to deal with the situation. Behind the scenes Obama's team and Paulson already were in contact to make sure the transition from the Obama to Bush would be well organized.
If a new crisis pops up it'll be up to a bunch of drunk circus clowns and the producer of the LEGO movie to save the global economy.