"If one considers the citizens of the Confederate States to have remained Americans..."
So you believe the only true "Americans" were those that fought for the Union? That the people of the Confederate states somehow became "non-Americans" and then when defeated conquered turned back into "Americans"?
This is an arrogant and insulting philosophy.
Once the war was over and the Union forces dissolved; after each seceding state was readmitted; the new United States, because it had grown and added more states, its Constitution changed, the federal military changed too. Today's Federal armed forces represents all of the states and their histories, including the Confederate armed forces and their veterans, not just the twenty-five odd states that called themselves the "Union".
The Confederate veteran was recognized by the United States Congress and the Department of Veteran Affairs as "Americans" long ago.
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David Upton