In that example, if the witness were able to communicate with whomever had the conn of the ship, it would be imperative to let that person know what was happening.
Okay, I chose a confusing example. In that one, the skipper would probably have nothing to do with the fact that his ship was being steered towards a big rock - some poor junior officer standing behind the shoulder of an enlisted steering wheel turner chap would be relying on incorrect data, and that
is where the alert bystander should direct his communications.
In the Trek case, Wesley would of course know that Kirk would be in actual charge. That's quite different from him merely knowing that Kirk was legally culpable, and would IMHO fully justify him yelling at Kirk when the Enterprise
As for Wesley being surprisingly happy about a technology that erases his own job, well, the air forces of the world today are full of people like that. They see great merit in taking the pilot out of the aircraft, and realize that this in no way reduces the force's need for gifted and experienced officers.
And that insignia of his? I don't believe it's "Starfleet Command" at all. It's just another one of those Fleet insignia. You know, arrowhead for 1st Fleet, starburst-flower for 3rd Fleet, rectangle for 11th Fleet, fancy lotus leaf for 23rd Logistics Fleet, etc.
But, malfunctioning off switches? Only in Hollywood. M-5 isn't a nuclear reactor: you can turn it off in a thousand different ways, some destructive, some not. And at least fifty of those will work come hell or high vacuum.
As regards taking these ships away from the front lines, I'm not sure if that's a big sacrifice at all. Perhaps Starships are predominantly exploration vessels with little or no use in defense (since the only "defense" job we ever saw Kirk handle was a purposeless vigil at the eternally peaceful Romulan Neutral Zone - and never mind that things unexpectedly went south there).