The Amnesty Act of 1872 removed most of the restrictions on former Confederates, save for "senators and Representatives of the Thirty-sixth and Thirty-seventh Congresses and officers in the judicial, military, and naval service of the United States, heads of Departments, and foreign ministers of the United States."

I would point out that the language in section 3 of the 14th amendment bars any person who had made an oath to the US before the Civil War who had resigned their commission or office to join the rebellion. Since Gordon had not held office or been an officer before 1861, he would have been exempt from that part of the law. Still, the Radical Reconstruction legislature(s) did find ways to eliminate CSA officers and politicians. As Ralph Eckert points out in his biography of Gordon "John Brown Gordon: Soldier, Southerner, American" (LSU Press) , the military governor of Georgia and other states in the 3rd district, Gen Meade, barred several Georgia candidates from the state constitutional convention due to their previous oaths. Meade allowed Gordon because he technically did not violate an oath, despite being one of the top commanding officers for the CSA at the Appomattox Surrender. He was a popular senator and Eckert makes the argument that despite his KKK connections, Gordon worked to unify the southern states back into the Union as long as Republicans and federal troops remained free from local and state politics. (Personally, I feel he is a bit of an apologist, but I get what he is trying to argue. He makes no attempt to whitewash the KKK aspect so that is a redeeming quality.)