When the government started making it clear that they would provide liquidity to large banks, they effectively extended their own credit rating to these institutions - for the same reason that Fannie Mae and Freddie Mac didn't fail because they were taken over by the government.
As noted, if there were more of a rapid bankruptcy structure where the government could have taken over a bank on a Friday to allow it to emerge Monday and restructure its assets, this could have provided greater confidence to the markets. The major issue of these institutions was that they weren't able to continue funding ongoing cash requirements.
Lehman went into bankruptcy because of liquidity - ie that their short term assets weren't able to cover their short term obligations. Much of this reason was because people didn't trust the valuations of their assets (e.g. subprime mortgages) and fears that proved to be overblown as to outstanding unstated liabilities - because it wasn't a transparent market. Barings, if you recall was brought down by a rogue trader so it wasn't beyond the realm of possibility and at that point the numbers being quoted as to the risk exposure of CDS's was in the tens if not hundreds of trillions of dollars (which again if netted out was only a couple trillion - and for that exposure to be realized would basically require every major company in the Fortune 1000 to go bankrupt at once).
How would they restructure their assets so quickly (over the weekend) and solve their liquidity problem without the Fed? (with the banks not lending to each other)
Even if a bank wanted to lend another bank money, it takes time to do a reevaluation (more than a weekend) of another banks assets and they would be more concerned of trying to calculate their own exposure.
The idea is that there wouldn't be a liquidity problem as the debt excluding individual depositors would be restructured as equity or some type of quasi equity/debt with assets aggressively written down.
The problem with what has currently been done with the bailouts is that the underlying problem hasn't be dealt with and only deferred meaning that at some point it's still going to have to be dealt with - and by taxpayers instead of investors who should bear responsibility for the risks that they have taken and the trust they placed in the executives of these institutions.
You don't think there would have been a run on the banks if more had failed and gone Chapter-11?