I know this post is 6 pages old but I wanted to chip in.
Two years ago my boyfriend and I adopted two kittens from a shelter (two brothers), we named them Chai & Chili.
When adopting from this shelter they had us sign a form that makes us oblige to have the adopted animals neutered when they're old enough.
We put a ton of love when we cared for them, two months in when they were 3 months the shelter's asked us to come to have them castrated, we were supposed to put them in the morning and take them home in the afternoon.
On that day, my boyfriend came along with a friend (to help him carry two cages), needless to say - he returned home with only one.
I remember the phone call, he called me to tell me Chili died (he was just laying in the cage, nobody even noticed he stopped breathing), at first I thought he was joking, how could a cat die from castration? The postmortem operation made it clear that there was a problem with his heart and it just stopped, it was unclear whether he was sensitive to the anesthetic (that's what they say at the shelter) or over-dosed (he was treated by a junior). I was MAD and utterly devastated.
Would I have done anything different if I had known it would end this way? Yes, I would've taken him to my veterinarian, or outright refuse to have him neutered. If it was a heart flaw or sensitivity then it probably would've killed him later in life, either way, I never regretted adopting him, even if my heart broke.
I loved him, he was happy. He had a good life, even if was really short.
After his death we adopted a new cat, so Chai wouldn't be alone (it was a hard thing to do, because we didn't want to just replace him, he was our child), her name is Cheetah. She is originally from the street, a week into her adoption we had to take her to the vet, she couldn't control her bladder. After the second appointment the vet noticed her heavy breathing and offered a scan, after the scan she was diagnosed with hiatus hernia, apparently she was involved in a car accident when she was on the streets. We were offered to take a chance and make her go under the knife and to repair her hernia. I was scared I'll lose another cat, when I agreed to pay 500$+ on a surgery for a kitten I owned for a week+ people were shocked, the vet himself was shocked because he said her chances were low. She survived, because I didn't give up on her, two years later and she lives happily with Chai in our household. The death of Chili still stings, even after two years, but it gave us the opportunity to save another cat, people kept saying how lucky she is that she got to us because other people would've given up on her.
The idea of not adopting an animal just because they were sick, it really bothered me, so I felt compelled to write this long-ish post.
It doesn't matter if they are dying, adopting them gives them a shot at saving them, and if not that, at least live happily with the time left to them.