I have really strong feelings on this as someone who owns very high prey drive dogs who absolutely will kill a strange cat who jumps into my yard. Funnily enough, my biggest cat killer was a fantastic search and rescue dog who saved many human lives, but boy would that dog grab a strange cat that jumped into our yard. He was great with our indoor cats but he did not generalize that to all cats.

I have spent thousands of dollars on fencing over the years to keep my dogs contained so I don't need to supervise them 24/7, but still got in a situation where my search dog killed a cat before I saw and could call him off. Tried to do the right thing and let the owners know since I knew where that cat lived, and they called animal control on me and straight-up lied about it happening on their property. I was pretty much saved by the fact that the scene was not clean and I hadn't cleaned up the yard yet, it was really clear it happened in my yard and within the bounds of my 7-foot privacy fence. Like what else y'all want me to do? My dogs aren't the ones roaming the neighborhood, and a high prey drive is really helpful in working dogs. (for the record, neither my dogs nor I faced any legal consequences beyond the investigation which found my dogs are safe and I was containing them appropriately, but it was still really stressful and if it ever happens again sorry but I'm just burying the cat in the garden)

Stomping a cat to death isn't okay, I'm not defending the neighbor, but I feel like a lot of people who own outdoor cats are kind of delusional when it comes to this. It's a dangerous way to keep a pet, there's a good chance they'll be chomped by a dog or hit by a car or killed by a wild predator or whatever. If you think the benefits outweigh the risk, whatever, you do you if you're in the UK or other places where cats have been around for ages (please no in many parts of the Americas and some other regions though, our ecosystems haven't been totally destroyed by them yet so we can still work to mitigate the damage), but I feel like you have to be prepared for this kind of thing if you have an outdoor cat.