That's really the only time goku cared and that's after he realized he kittened up having Gohan fight cell as piccolo pointed out Gohan isn't like goku and doesn't like fighting.When Master Roshi almost died during the Tournament of Power as said before, as well as when Krillin died at the hands of Tamborine when King Piccolo made his move. And also technically when Frieza wounded Piccolo then killed Krillin in front of him to go Super Saiyan. Goku just needed a lot more development, but we didn't see much of it as we did. But we saw too even for his flaws, he loves his family so much, considering his willingness to sacrifice himself when Cell was going to self-destruct when Gohan just toyed with him when he went Super Saiyan 2 the first time, as well as the anime having him kiss Chichi good-bye before he went to prepare to go to the Hyperbolic Time Chamber to train with Gohan before the Cell Games, as well as his interaction with Goten when he first met him.
Yeah, that always pissed me off too. We see it too when Sasuke sees the destructive power the Rasengan has, even if it isn't meant to fully kill an opponent like the Chidori, BUT the force of the blast from the release of the chakra can kill someone. The part that's also frustrating about that whole thing, on top of what i described, is how Kakashi made it seem worse that Naruto learned a Jutsu that HIS OWN FATHER created and said it was too soon for him to have it, but it's A-OK to teach Sasuke something like the Chidori when he knew Sasuke was mentally unstable due to his superiority-inferiority complex as well as his desire for revenge.
Again kakashi sucks this could have all been avoided if he actually acted like a teacher and got Sasuke the help with curse mark and his own mental issues or pulled rank on him and put him in his place.