A bald-face lie. The Confederate States never sought to, or attempted to, overthrow the United States government. They sought independence from that government. Even the term "Civil War" is a lie, as the CSA were in rebellion and soufght independence, as the 13 colonies had done not 100 years before.
True enough, but there's no telling what the reaction of the Confederate government would have been had the Army of Virginia been successful at overrunning Washington and capturing Lincoln. Having bowed to the temptation to institute conscription it's difficult to consider the possibility that had the Confederacy been given the opportunity to not only remove the possibility of a future threat to their northern border but to return the former states of the Union to a governmental system within the bounds delineated by the constitution, they might well have done so.
Alternately, the possibility that at least some of those conquered territories might have been traded to England in return for the diplomatic recognition and military support sought by the Richmond government, particularly along the Canadian border, is at least within the realm of possibility, and would have left the Yankees less able to later stage a future retaliatory war against their Southron neighbors.
But of course their act was, as you note, not treasonous but rebellious. It was Lincoln whose actions were acts of treason to the constitution and gave those seeking the dissolution of the Union the moral justification for their actions.
-archy-/-
.....that's quite true.....in a civil war you have two [or more] factions each trying to sieze control of the government....the "late unpleasantness" of 1861-65 was a war of secession & partition; a concept as American as apple pie......
FRee dixie,sw