Additionally, because of the command structure of the Confederate Army, all chains ran through him. Technically, Lee (not being the CIC of the Confederate Army, but merely the Commander of the Army of Northern Virginia ... one among "equals") had to go through the Department of War (and, as a result, Davis) for anything dealing between areas of jurisdiction. Davis would have been much better used by the cause of the Confederacy by being the Secretary of War (or even, with his military background, a general over one of the armies) because that's where his true interest lay.
There was not an inch of compromise in his soul, not a millimeter of bend in attitudes and beliefs that had had. This, in the end, does not make for a very smooth-functioning administration: his Vice President hated him; many of the State governors refused to get along with him or support him in the overall war effort; his relationship with many generals in the Army (particularly Joe Johnston) was atrocious.
A different President for the Confederacy may have made an extreme difference in the progress .. if not the outcome .. of the War.
Not that Lincoln's relationship with his own generals was much better during most of the war, but you're right. This is something frequently lost in more romantic analysis of the war.
If I recall, after losing command of the main Confederate Army around Richmond (soon to gain renown as The Army of Northern Virginia under General Robert E. Lee) due to a bullet wound, Joe Johnston bitterly commented that the bullet that struck him was the most fortunate thing that could have happened to the Confederacy. Not that Johnston thought Lee was a better general than himself. He didn't think so at all. But he noted, accurately, that Lee would be more effective because he had the confidence of the president (Jefferson Davis) while he did not.