I agree with this rule.
As a child, every time I encountered a character that was cruel to animals, I was like come the fuck on, that person is already clearly bad, how fucking lazy is this writer?
There was one book where there was one character which was cruel to cats in a very specific way. In 34, I am still feeling traumatized by that short, textual description.
I'm not disturbed easily. Just to quantify: a few years ago, I've seen an aftermath of a fatal car crash: bloodied children, first responders doing CPR on mother's body that didn't need it at that point (she was a goner; news said, she didn't wear a seatbelt). I helped clean the debris for the ambulance to come through.
I was fine at the scene. It wasn't on my mind a eek later. I didn't forget, but it wasn't disturbing anymore. And I didn't even know one can flip an SUV in clear weather.
But I can't get over that fucking cat murderer. I don't even remember the name of the book, or anything else from it. Just the way he did it.
I gained absolutely nothing from that, except a strong hatred for this cheap way of creating a villain.
Even though I learned, when I grew up, that some people do torture animals, and that it is a predictor for other horrible things that they do. That book might have been not just realistic in that description, but outright real.
..I still wish I could've un-read that part.
The only other thing that affected me that strongly was watching "Dancer in the Dark" as a teenager. It's still one of the movies that I'd rank as the Best of All Time. I have not seen it twice, and I don't feel I'm ready.
But at least the cruelty wasn't pointless there. It was a means, not the point.
Cruelty to animals as a villain's trait is rarely a means to an end. A hunter that skins his game isn't an example of that.
But, say, a character that skins an animal for fun is traumatizing.
The problem here is forcing the child to choose between being unempathic or traumatized.
Because either you empathize with the defenseless, tortured being (traumatic), or you get desensitized and learn to not feel bad about cruelty for cruelty's sake.
Which is precisely what we want to avoid.