There is a s difference between being in the field and being on the front lines. However, as I said, they cannot be armchair generals.
Again, you're deliberately misinterpreting what I've said. A battle-wizard does not have to be a simple blasting gun, but they must still be nearby to reasonably effect the outcome. There are many many more ways an extremely powerful wizard is of use if they're in the theater of war (this could even be as far several kilometres away from the battlefield, but still close enough to be called in with good time), than if they're cozy in the capital having tea with the lazy nobles back in Altdorf.
Also remember, all of the criteria you have suggested so far imply a politician as opposed to a leader of a fighting force (yes, politics is often involved in higher military matters, but you can look how well emphasizing that as a quality to seek turns out in various militaries across the world). The Empire remains especially leery of wizards getting involved in upper level politics in the empire, for good reasons given the grey wizards repertoire.