Most of the field armies in the Civil War were under the command of guys who basically didn't listen to any grander strategy, especially in the Confederacy and early Union.

Lack of western European aid and fall of New Orleans prove Southern independence movement not achieve

>Why were there no serious Confederate attempts to relieve Vicksburg?
Lee was too far away for his forces to make it to Vicksburg in time once the situation became dire enough, Bragg attempted to send some amount of troops iirc but he had barely any men to spare because of the threat of Rosecrans' Army, and Confederate forces west of the Mississippi were either unable to cross the Mississippi or simply refused.
>Isn't the protection of a transport artery more important than capturing the enemy's capital?
Yes and also no in a way. IMO I think the defense of the Mississippi was a fruitless effort after the fall of New Orleans and Corinth (which by proxy made Memphis entirely untenable) since the Confederates couldn't protect the Mississippi long term after the fall of those positions. Definitely too vital to simply abandon, but resources could have been better allocated and letting such a large amount of troops get bottled up in a doomed siege was stupid.
>Even if the Confederates won at Gettysburg Lincoln could have just relocated.
More likely the Confederates wouldn't have been able to threaten D.C. even if they win at Gettysburg unless it's a victory of the same scale as 2nd Manassas or Chancellorsville.