After fending off most challenges to its independence and winning new powers to oversee big financial firms, the Federal Reserve has emerged from a bruising debate on the overhaul of U.S. financial rules as perhaps the pre-eminent regulator in the sector. But that could only bring it added blame if things go wrong again.

Just a few months ago, amid populist anger at the Fed for failing to prevent the financial crisis of 2008 and bailing out Wall Street, Congress was talking of stripping the central bank of its supervisory...