Like how he doesn’t give a shit his son died

Did you watch the film? When he hears his son died, he falls by a chair. Then he keeps saying to the Klingon commander "you bastard, you killed my son" several times. These actions do show he did care his son died. It appears you did not see the film.

You obviously missed this one excharge between McCoy and Kirk...



[Witnessing the destruction of the Enterprise]

Kirk : My God, Bones... what have I done?

McCoy : What you had to do, what you always do. Turn death into a fighting chance to live.

I don't think the words "doesn’t give a shit " mean what you think.

I'd say he very much gave a shit, he responds with shock and grief here, and he even holds a grudge with the Klingons that is exploited by the villains in Star Trek VI!

He cared more about Spock, but to be fair, Kirk didn't have a relationship with his kid.