He doesn't need to be. Read the clause again.
"No person shall be a Senator or Representative in Congress, or elector of President and Vice-President, or hold any office, civil or military, under the United States, or under any state, who, having previously taken an oath, as a member of Congress, or as an officer of the United States, or as a member of any State legislature, or as an executive or judicial officer of any State, to support the Constitution of the United States, shall have engaged in insurrection or rebellion against the same, or given aid or comfort to the enemies thereof But Congress may by a vote of two-thirds of each House, remove such disability."
Merely engaging in the insurrection is enough to disqualify you.
Now, there was a practical reason for this at the time. Putting every leader of the Confederacy on trial for treason would have opened a lot of wounds and caused a lot of trouble. So, no, not even traitors like Davis and Lee were put on trial. Which is too bad, they should have been, and we wouldn't be having these stupid debates today about Confederate Flags and Statues.
It's why you don't see Nazis in Germany today. We put the Nazis on Trial. We hanged the worst of them.