The Confederate government was a legitimate government and did its job, serve its people. What many historians focus on is the international recognition as a government. This is important in determining legitimacy in relation to world politics. The Union warned England that they would go to war if the Confederacy was recognized, (but they did an odd thing, they also demanded that England respect the Southern people as if they were still in the Union).

France and England did not recognize the Confederacy as a government but they came very close. France and England applied international laws and declared the Confederacy as a "belligerent" giving that government equal weight to the Union in respect to the war. So the Confederate Government was a working government even if it did not get the final recognition from other countries.

One could argue that Lincoln's Government was not a legitimate government over the seceeding states, because of his minority status in the election and the inability to enforce the laws and protect the citizens in states and territories in the Confederacy.
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David Upton