> liquidate the failing companies, use that money

What money are you expecting to get int hat scenario? If I had to guess it wouldn't be a fraction of the cost.

> buy out stakes of pension and index funds

Index funds are part of the reason the top companies stock gets so bloated compared to those just outsid of their frame.

>If any important bank fails because of this, nationalize it and let their customers exit using money from the federal reserve. Then try and recover as much of that as possible using the banks investment portfolio. Costs a lot less than bailouts.

I agree. It worked here in Belgium tho they did spin off toxic elements of the bank in a not so great way prior to doing this.(some regular people still got shafted)