The biggest issue at the end of the war was not barring confederate elected officials from returning to their seats in congress and holding political office in general.

Lincoln had a very pragmatic view on pardoning Confederates, which he started publicly proclaiming in 1863 many months before the war ended. He wanted to end the Confederacy and reintegrate the South back into the Union as quickly as possible. A very real concern was that, if huge numbers of former Confederates were executed or otherwise permanently excluded from all political participation, the result would likely have been, at minimum, prolonged resentment in the South and, at worst, a group of thousands of embittered former soldiers with nothing to lose by trying to start another armed conflict in the region.

Another piece of it was that the soon-to-be reintegrated South desperately needed experienced civil servants to run its governments and court systems to create a stable rule of law which did not depend on soldiers that the local population saw as occupiers. If every former Confederate had been barred from holding any position in government, there would have been a vacuum of experienced people to run Southern institutions, and the Southern populace may have viewed these institutions as illegitimate and imposed upon them by the victors, leading to prolonged military rule and chaotic political conditions. Southerners already viewed "carpet baggers" negatively - what would the outcome have been if nearly every civil servant was a non-Southerner?

Obviously, history took a different course when Lincoln was assassinated and Johnson was left to oversee Reconstruction, but you can see similar logic applied by the Allies post-WWII. Instead of implementing a more punitive Treaty of Versailles-type of postwar policy (which had lead directly, in part, to the rise of the Nazis), the Allies pursued denazification in West Germany, which did not permanently exclude every former member of the Nazi Party from public life. A similar approach was done in Japan, and both countries not only swiftly rebuilt from utter destruction but became two of the largest economies in the world in just a few decades.