You'd rid the world of a LOT of dogs if you wanted all the ones that would grab and harm or kill a cat euthanised, pretty much anything from a jack russell up might be inclined to do that, and breed is not the only factor involved. I've had terriers and hounds that would not and known gundogs and companion types that would.
I've also had sighthounds that were perfectly safe with their own cats but absolutely not safe with 'other' cats, and whilst it is possible to train them to ignore cats, it's not possible to train cats out and about not to ambush dogs from under cars and hedges - if you've never had to try and peel a furious cat from the wire muzzle and skin of a terrified greyhound, whilst one yowls and tries to deflesh and the other screams and tries to run, you won't understand what a hazard they can be!
Most of the loose 'greyhounds' running around fields in my area are not lost rescue pets who failed at recall with idiotic owners, but intentionally dumped coursing dogs, or dumped stolen dogs. Of course its very easy to lay the blame at the feet of rescue dog owners and there will always be the odd idiot, but they are not the main source of straying sighthounds in most rural areas!