I just play it out in my mind, with history as my guide.
At its most basic roots, the Fed was created to facilitate fractional reserve lending, in exchange for the ability of government to engage in deficit spending, as both siphon wealth from the same trough.
Bailout is another word for "allow defaults without recourse". So that's really the question, or at least the first part of the question:
What would allow the Fed to default without recourse? The second part make it a complex question, with the qualifier "...while allowing cover for fractional reserve lending and expansion of the debt-based money supply by commerce, while simultaneously allowing government to siphon from the same money supply by borrowing against the productivity of society on the whole?"
The answer becomes as simple as the solution that is always proposed; namely, more of the hair of the dog, only an expanded scale. That means an even more centralized facilitator, one that has the license to revalue and restructure all debts, public and private, while providing badly needed "elasticity" AND "liquidity" (read=counterfeiting by any other name) in "times of crisis" (read=at all times, as a matter of force).
What the Fed did for national banks, a "larger Fed" can do for multiple Feds.
In other words, the very last and ultimate Ponzi scheme treadmill, which could last, in theory, for hundreds of years, before it finally mega-tanks, as all infinitely expanding Ponzi schemes must - after which (and many, many generations after we are all long dead and forgotten) the entire world becomes third world, divided completely into rich and poor with no in-betweens, before it is finally forced to divide, like the Soviet Union, once it was clear that there really is nothing left to siphon.