But this adds absolutely nothing to the film and actually bogs it down. Plus, then you'd have people bitching about how starships don't work that way.
Sure it does. It gives the Enterprise a moment where it actually did something to stop the Vengeance.
To suddenly make it a rescue mission would shift the tone to a more hopeful one and I think this sudden shift would damage the movie
Isn't that what we get in the film anyways? We have a tragic, quiet death of Kirk followed immediately by another action set piece with Spock trying to kill Khan out of vengeance which than turns into a rescue mission to save Kirk. If you think having two shifts in tone is too much, I'll understand that better. But don't say that shifting tone on it's own is a bad thing, because that's exactly what the film did.