It is odd that Kirk never brought him up, and it does make it really obvious that this is an invention for the movie.
I think it works because it ties into the main themes that the movie presents on growing old and facing responsibilities. Throughout the film, Kirk is struggling with facing death and all other "no-win" scenarios. He's managed to dodge his way around consequences, whether as a cocksure captain or a too-clever-for-his-own-good cadet re-wiring the Kobyashi-Maru test. Wrath of Khan (and Bone) gives him reading glasses, a family that he can't be with (they're estranged), and the inability to save all of his crew (Scotty with that kid is just the beginning...) So, Kirk needing to deal with David and Carol not wanting him around - that's part of this "grow up and face the music" idea that makes Wrath of Khan so great.
In terms of a possible in-universe explanation - and again, I acknowledge that any explanation is doing the movie a favour because David is a clear invention - I think we can have a couple of good points.
1) Kirk might have spoken about David and Carol to his crew, we just never witnessed it. This is kinda weak. While it is believable (we have ~70 episodes of Star Trek - we didn't see every conversation Kirk had with his crew over five years) it's bad writing to rely on something like this.
2) Kirk might have learned about David long after leaving Carol. Kirk had many affairs in the stars and, often because of circumstance, moved on from past loves. He might not have even known he had a son for years. I don't remember if there's dialogue around this in Wrath of Khan, so forgive me if this is canon (or contradicted) in the film. If Kirk was kept in the dark about David for ages, it might be the case that he didn't talk about this relationship because it hurt him. The very quality we love about Kirk, that you've pointed out - his care for his family, friends, and crew - would make being shut out by his lover and son particularly painful.
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