I don't think it has anything to do with logic. It's laziness and a lack of willingness to spend money. My roommate in NYC worked for a vet clinic that was famous for rescuing animals that other vets would turn down, and seeing they had a full recovery and a healthy life. Imagine if we just gave up on people that easily. And for a lot of people, animals provide a comfort and love that they can't have with other people. Especially the elderly.
it's like, say I had a family member I fucking hated, and they were like 'well, we can let them die of this ailment or there's a 60% chance of recovery with the proper surgery, if you want to pay 10 grand.' If I were super shitty or totally broke, I'd be like 'fuck it'. But if you said it about someone or an animal I loved, yeah, I would fucking do everything for it.