Killing the first one probably won't any more than killing a single mouse doesn't stop an infestation, but once the mice realize that you got a couple of cats they will find the exit.
That's why you need to put traps by the exit too, the mice know too much and might potentially regroup to return with better tactics. You've got to get them all in one swift purge.
Wouldn't it make more sense to take their money than their lives? Use the pheromones to make the billionaire transfer all his assets to groups fighting climate change.
The thing is that we've gone through multiple generations of new owners each thinking a bit more than the last. Now they know how to appear decent while continuing to exploit earth, humans and other animals.