I talked before about how Papyrus’ inability to kill the Player - or rather his ability to always reduce the Player to just one HP - is an indication of not just his incredible kind-heartedness but also some remarkable skill. Considering no other character in the game can safely reduce an enemy to just a single Hit Point with no risk of accidentally killing them. Including other powerful characters who would have an obvious incentive to do so.
So I think that another factor in why Papyrus is the Best at Not Killing Anyone (outside of his general moral disposition and dedication, which are also important to note) is… probably Sans. If Papyrus ever had anyone else to spar/practice FIGHTing with, it would only be his brother. (Remember, Undyne’s ‘Warrior Training’ is actually just cooking lessons). And Sans only has one HP - Papyrus had to learn to be careful to make sure he never hurts his beloved older bro too much.
In contrast, when Toriel had to learn to use her magic, that was probably before the War of Humans and Monsters, with her old Boss Monster family? Or at the very least with another Monster that is more powerful than ‘the weakest enemy’. So, even if she never ever wanted to use her powers to kill anymore - she never had to learn to be as gentle as Papyrus had to be gentle. I mean, she had some skills at weakening her attacks and making them safer, but they are not quite as precise as Papyrus’ is. And so…