The West was the gateway/ choke point for the economy and physical separation of the South. Cutting the Mississippi cut the South off from Texas, Arkansas and Louisiana. While taking New Orleans, Memphis, Vicksburg etc cut off its commerce. Plus the West served as the invasion avenue into the deep South and what industry the South actually had.
I'd wager they had equal roles. The CSA had to protect its capitol, but the west siphoned off troops. The CSA also had to protect its middle, but the ANV starved the rest of the confederacy for material.
I'd also point out the 3rd theater. The South Coast and the Blockade. Winfield Scott deserves a lot of the credit for beating the CSA. His Anaconda Plan proved to be a war winner. The CSA with its much smaller resource base simply could not fight on one front and win a decisive victory because they were being squeezed on three front, losing any of which would cost them the war eventually.