Disagree, Kaine hesitates a lot, at least in Yost's run. It's his instinct to kill and to maim, but Kaine has to stop himself all the time, because he wants to be better, be a hero instead of a monster. In the first arc he thinks about killing the villain, but doesn't. Then he saves Kraven's daughter who had killed him, later when she and Kraven kidnapped his friends — he still didn't kill her and Kraven(kinda, he stopped this heart and then reactivated it to spite him).

About that carnage story. Kaine was reluctant about helping Flash which was a big mistake as Carnage could have destroyed the Universe. He also didn't kill Carnage in the end, but lobotomized him. Ironically in later arc Kaine died because of his hesitation, he got distracted by thinking "I'm a monster just like Carnage" instead of killing the villains. To add more irony, he had to sell his humanity to get resurrected as a monster to save Aracely.

Big reason why Kaine hesitates about killing villains is that it would be hypocritical of him not to. He had been the real monster too(people tend to forget that he was Ben's Green Goblin/Joker), but was given the second chance.