Why doesn't the fed give the money to the people unable to repay their loans instead of propping up their balance sheets with it? Would that fix the solvency problem? Probably not, right, because you can't borrow yourself out of a debt problem.
Heh, that's probably not too far from the truth.
This had been attempted early in the crisis - $500-600 was provided to each taxpayer. The problem from a bank/gov't perspective is that there's no way to guarantee that money will go to consumer spending (or the right kind of spending), thereby supporting economic activity. If it goes toward paying debt down, that exacerbates the banks' condition (the loan may be paid off, but there's still a cashflow shortage). Essentially, attempting to herd cats is much more difficult than just putting the funds into banks where it can then be more effectively directed.
At least, that's the gist of the reasoning. In reality, the banks are black holes due to their insolvency and are dragging the entire system down - but because there's no external reference point, it's hard to see what's actually going on. You could picture a family in a house, with each family member representing one of the major world regions. All of a sudden, a sinkhole opens up under the house and it goes into freefall. Unless one of them looks outside, it will just seem like someone forgot to pay the gravity bill, so they try tying weights on themselves to counteract their floating.
Now it's a catch-22 where immediate collapse of the banking system as it is today will cause massive disruption because of the extent of dependence upon finance, and continuation of the banks' existence by supplying liquidity is just making the problem even bigger without actually fixing the underlying cause. The central and supranational banks (BIS, IMF, etc) have to decide which course of action will be most effective, and that's direct liquidity infusions into banks. It's like the difference between inhaling a drug or injecting it when you've got a plastic bag over your head and can't breathe.