Kirk is out of character

Does it not bother anyone else that the idea of Kirk having a secret son, who he never sees and is never mentioned in TOS, seems completely out of character? He overjoyed when Miramanee was pregnant with his child. In the three other instances where he truly fell in love, Edith, Elaan, and Rayna, he wanted to be with them long term but couldn't. He cared about his brother and nephew, not to mention the strong bonds he had with Spock and McCoy. But not his own son? And keep in mind TOS Kirk did not spend all his time chasing women like alternate Kirk (instead only using it in escape plans).

I could understand a secret son from tight-lipped Spock, and indeed we do later find out that Spock had a half brother and adopted sister (and a secret son in the beta canon). McCoy had a daughter in the writer's guide but this was never made canon, and considering McCoy's divorce, it doesn't seem out of character that he might not wish to discuss this.

Kirk's son is the most grievous example out of all the films but I find overall that TOS Kirk and movie Kirk to have a different feel. Movie Kirk is basically Shatner in STV but Shatner also creeps into the other films too. I don't know if the secret son thing is necessarily Shatner but it's certainly not TOS Kirk.