Because life is priceless, and the life of the cat you care about should be priceless to you. When you get a companion animal, that’s what you sign up for. And if you’re not willing to go the distance to take care of your pet when they are sick, you shouldn’t get a pet.
There are so many episodes in all the series but here’s a few from Voyager: VOY: “The Chute”, “Dreadnought”, “Learning Curve”, “Meld”, “One”, "Once Upon a Time”, “Timeless”… the list goes on. Many other episodes focus on a single member of the crew, many times with the Captain not being an important part of the story at all.