For some jws who've put off kids in 'this system' to 'do more for Jehovah' and for whom the end hasn't come, their pets are all they've got. And the org wants to take that from them.
Whoever thinks pets have no value has obviously had no experience with them. They provide companionship, love, affection, entertainment, interest, calming and so much more, and all they ask back is your love and attention. To deprive people of the joys animals can bring is heartless and ignorant.
Case in point: I used to breed quails. Tending my flock and raising beautiful birds from eggs to little fluffy chicks to watching as new plumage patterns emerged gave me intense satisfaction. Even watching them dust bathing like mad in a new tray of beach sand was delighting. I was understandably proud of my quails and jws at the hall often asked me how they were or wanted to see them. So of course I'm telling the co about the joys of my quails and he said, "wow, one day I'm going to hear that you've given up your quails for Jehovah and joined the full time work!" Like I was somehow less spiritual because I loved birds. I also told some elders about how much joy the birds brought me that Jehovah wouldn't want to take away something that made me so happy. I'd been suffering from depression and anxiety cuz of being a jw and breeding quails was my only outlet. Point is, I was not a good association because I took time to care for 50 birds.
That seems to be the jw way, unless it's an elder's wife. She can worship her cat but no one else can spend time with a cat, a dog or a bird.