I think it is even simpler than that, after all the reach of airpower means that we can - sometimes - stay at home and still provide effect to a joint commander (obviously not to the same scale as the B2 force though).
Warfighter first is a mindset. It means that we are primarily airmen and RAF officers; therefore, our profession should be the employment of air power. We should all have a level of understanding of what this entails which is, in its simplest form, warfighting from the air (and space for any FASOC junkies). Being an infantry soldier is a specialism, as is being a techie, chef, mover, air traffic controller or even a Harrier pilot. So a gunner (like any specialist) needs to appreciate both that he is primarily involved in the prosecution of air power and what that entails and that he uses both his basic Service skills as an AIRman and his specialist skills as a gunner to achieve it.
This is the same as the Army's 'soldier first' given that that organization's purpose is Land combat (with a little bit of air) - or 'soldiering'.