Graphite is used as an anode material because the redox potential for ion intercalation is low.

Carbon material is used in cathodes to increase conductivity, but ideally has no contribution to capacity, perhaps a negligible amount from surface absorption. You could add graphite to the cathode, but the redox potential of your active material is much higher, intercalation into graphite does not occur until low voltages, perhaps low enough to irreversibly change the active cathode material.

When anode materials, such as graphite, are tested in a metal half-cell, they essentially act as the cathode with very low voltage relative to the metal anode.