I think you should research the process of bankruptcy (and the Great Depression) before proposing solutions. Bankruptcy is not evil and does not destroy wealth -- it shoves bad companies aside and makes room for those who stand the best chance of making use of the assets that remain.
It sounds like you're proposing what Japan has been doing for the past decade. Look up "Zombie Corporations Japan" to find out what ultimately happens when you back bad companies with taxpayer dollars to keep them from going bust in order to retain some daft idea of social harmony.
A capitalist system isn't born or sustained by government-managed compromise kicking in when your favourite team loses.
Of course they have been. That's the core of the problem. Did you even read my post before replying?
This discourages companies, particularly banks, from trusting each other.
The problem we're having now, which is also the problem that brought the Great Depression, is lack of trust. Banks won't lend to each other, or to other businesses, because they don't trust that they'll get their money back.
Saying that we should let the banks go bankrupt at their leisure is just letting this problem of trust fester for the next few years and resolve itself one bank at a time, slowly. While the problem is correcting itself, the market doesn't allocate money where it's needed because of the lack of trust, and everyone suffers. The total cost of that is much larger than a trillion dollars.
At the other extreme, you could just nationalise the whole banking system. That's a better option, but still not ideal, because governments suck at running businesses, and this would also be less efficient in allocating capital where it's needed (the primary function of banks),
So, somewhere in the middle lies the compromise. How can we get the banks to start trusting each other again and yet not nationalise them? Well, one way is to offer them a kind of "knife armistice". Bring out your knives to the police station, hand them in, and we'll let you go free. We'll even give you a water pistol as a compensation prize.
Don't be fooled into thinking that just because the bailout bill doesn't put the CEOs of banks in prison, that it will not be followed by further regulation. For better of for worse, tighter regulation will be on its way. But that's a separate debate that's not tied to the bailout. The purpose of the bailout bill is to resolve a crisis right now, not to make long term fixes to the system.
Banks are special types of companies because of their effect on the economy. You might let Enron go bust, because after all that's unlikely to cause widespread damage to the entire world economy. Let the banks go bust slowly and painfully, though, and every business pays for it. Calling for "Bankruptcy not bailout" is terribly short-sighted.