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?
Why? Why didn't you tell me?
How can you ask me that? Were we
together? Where we going to be?
You had your world and I had mine.
I wanted him in mine, not chasing
through the universe with his father.
But you didn't post what Kirk said to Carol after David attacked him.
Kirk (dodges attack and looks at Carol): THAT'S David?!
He knew who David was. He just hadn't met him (probably since he was a baby etc).
That's my take and I'm sticking to it