Well first we need to ask why we even bother with independence and credibility for central banks. The simple answer is time inconsistency – if a central bank wasn’t independent from government it would not be able to stop it self pushing for seniorage, which would then be priced in by households and firms, which would lead to straight out worse outcomes!
Now this never, ever, meant that fiscal and monetary policy shouldn’t co-ordinate. The two have an impact on each other, and there is constant communication between those two parts of government. But by giving a central bank independence, it can gain credibility when it comes to setting its “time path of policy”. A central bank that is independent won’t have an incentive to transfer resources to fiscal authorities, and so will gain credibility with the public regarding this.
During the crisis, the actions of central banks – and the fact that many of them have had to take on large amounts of government bonds, and even mortgage/business debt, has blurred the line regarding their independence. However, I take the view of the paper I just linked to in the previous sentence – in truth as long as central banks can reverse these positions as the economy recovers they regain their credibility. In truth, during a massive financial crisis like the one we’ve experienced you WANT your fiscal, financial, and monetary policy authorities working together.
Where the financial crisis came from
If we believe that the financial crisis stems from there being a LOLR that decided not to be a LOLR (so we had a build up of debt based on moral hazard and lax regulation, followed by a messy bank run once it was unclear whether there was a lender of last resort), then the crisis occurred because the Fed lost credibility on this front (followed by the ECB). If anything, the crisis suggests that the rules and policies of central banks need to be more transparent, and their operation kept more independent from the greasy hands of governments that want to use a stealth inflation tax – it doesn’t suggest that we now need to throw away independence completely.