That argument is seriously flawed. Paul says that Congress does not have the power to delegate authority to coin currency, but he runs into a little trouble farther down in Section 8:
"The Congress shall have power...To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers"
Congress deemed it necessary and proper for a central bank to assume the duties of coining currency and controlling monetary policy. By default, Congress still has the power to control these things, all they have to do is pass a law abolishing the Federal Reserve. And that's exactly what Paul tried to do. Essentially, he defeated his own argument.
Of course, this is just laying aside whether or not it would be practical to return to the old system. It's not his flawed argument that makes him nuts, it's his archaic view of international economics. History has shown again and again that nations with independent central banking systems have more stable business cycles, and are far less susceptible to panics. Before the Federal Reserve was created, there was nothing to stop a run on banks from escalating into a full blown financial calamity. The lack of a lender of last resort resulted in several significant panics in the late 19th and early 20th centuries. The Federal Reserve isn't some crazy liberal theory, it was an institution that was created after decades of painful trial and error. To revert back to the old system, especially in an era of instantaneous banking and global interaction, is nothing less than pure lunacy.