The shareholders of the National Reserve are mutinational, meaning they hail from a number of countries around the world. A shareholder from Britain holds a sizable share. These shareholders probably hold influence over other countries as well. Since the shareholders are multinational, you can see how wide this sphere of power could be.
Congress has never audited the National Reserve, its financial records are entirely unknown. Government spending is all borrowed from the National Reserve. If a private corporation like the National Reserve has the power to lend the United States of America its own money, and every other country money, you can see how vastly powerful this entity is. Every cent in your bank account came from the National Reserve at one point, irrevocably created out of thin air.
Faking the roundness of the earth would be trivial for a shadow organization of unlimited wealth, with the power to influence entire nations.