Is that independence really going to be there in time of war? I haven't been stationed on a CRUDES but what I have seen from a CG CIC in action is the TAO reporting to the force TAO, who didn't do anything without AW's approval. And that has been for exercises and training missions. It's only going to get worse from there.
Depends on what was going on, and what you consider "anything."
Real world example:
MASON playing chicken with the Houthis, pretty good odds at least one of those engagements was entirely driven by the TAO...CO simply can't be there 24/7 and things move too quickly.
Obviously, when we step up to integrated CSG ops, things start to change. But self defense employment doesn't change, and for the rest, it's more about how much AW wants to micromanage the problem. That and geopolitical context.
Although that highlights another point...AW staff is for the most part just the CG's crew. Which increases the level of responsibility for the TAO and AAWC JO's on that ship.
And if the W ship does down, backup W ship is usually a DDG....which means all that responsibility to manage the air battle just got piled on another JO.
But we all work for somebody else when it comes to the blue water CSG fight.