If you add Kirk/Carol too this will become the Kirk's show and him and Spock won't be equal as protagonists anymore and honestly I'm not interested about a Kirk show. My biggest problem with this movie is how Kirk centered it was. Been there, done that.
The Orci/Kurtzman Trek will always be Kirk-centered, though. IMO Bad Robot sports a serious hard on for the dude.
Besides, I never liked the original Carol and I never thought that the Kirk/Carol thing was supposed to be this big love story that we absolutely need to see again in another reality. I mean, they have a fling when they're young and they have a son but Carol denies Kirk the chance to know him so kirk doesn't even know he exists he gets to know that only many years later when he's old and it's too late to be a father.
I think Carol did tell him that David was his long before TWOK. I didn't get the idea from TWOK that she never told Kirk until the events of TWOK. Where did you get that impression?
I have to add that reading the after darkness comic is sort of confirming my worries about her character and how she seems forced.
putting under the spoiler tag
Example: Spock is in the midst of pon farr and Uhura takes him to sickbay where he wakes up to find Uhura, McCoy and Kirk looking over him. That's his girlfriend, the captain and the doctor who also are his friends so yeah it makes sense for them to be there. There is no reason for Carol to be there too, though, and away from her post. Yet, for some reason they put her in the scene which is a bit gross because Spock has to explain what is happening to him and you know how vulcans don't like to talk about pon farr especially in front of strangers. Was it necessary to have her hear it? No, but Spock's problem is essentially used as an excuse to give to Carol a little interaction with Kirk even if brief.
Later when they decide to take Spock to new vulcan, Carol asks Kirk the favor if she can go with them using what I honestly think is a pretext from a writing pov. It's further obvious when you compare her to Uhura who is, after all, Spock's girlfriend, and she has a valid reason to go with him and yet doesn't feel asking Kirk, it's the captain that understands on his own and asks her to join them in the shuttle (Kirk gets points for being discreet)
I wish they had introduced her from the start because now I can't help but find it pretentious and forced, I mean, it's a tiny bit blatant
Hmm... yes, it's worrisome. Ugh.