Very selfish n unlovin behavior...Yes. He does.
I was going to answer his other message thread until I saw this one.
No, thyroid issues are absolutely not a good reason to euthanize a cat.
It's not like kidney failure or cancer or something like that. Cats can live with thyroid problems, just fine, if they are taken care of. In some cases, thyroid problems are transitory, meaning that they seem to go away by themselves.
Read here: Thyroid Hormone Deficiency in Cats | petMD
In my opinion, hypothyroid is just one of those things that happens to cats when they get old. It's a natural part of the aging process. This is how Mother Nature tells us that your cat is getting ready to pass on to the next life.
Your job as a cat's main human is to help her transition to the next life by making her as comfortable as possible until the time comes.
If the cat is in pain or otherwise suffering it MIGHT be reasonable to euthanize but, in this case, the cat is likely NOT suffering. She's just growing old.
O.P. mentioned that the cat lives out of a garage or something like that. Right?
What about making a nice place in the garage for the cat to stay where she can be warm, comfortable and well fed until she comes to the end of her road? Make her a nice bed to sleep in. Give her a "kitty condo" to live in. A clean box with the front cut out like a miniature dog house would fit the bill, nicely. Put some of her favorite food and some clean water nearby.
You should want your cat to know that she was loved and cared for in her old age before she passes on.
You SHOULDN'T want to uproot your cat from her home, throw her in a cage at some animal shelter until she dies or put her to sleep in some sterile hospital room. That's heartless!
Don't forget your mother. She probably understands that the cat is old and is coming to the end of her road. She might even understand that thyroid problems are just a symptom of aging. Maybe she's already trying to make the cat comfortable until she transitions.
I don't see any evidence that O.P. has even considered that. It only seems like he's fixated on taking his mother's cat away and killing her. Put the cat's issues aside for a moment and ask yourself, "Do you really want to hurt your mother like that?"
It's her cat. She's probably known this cat for a decade or more. It's been your mother's friend and companion for all that time. Now, you're planning to throw it all away?
That's some pretty selfish behavior.
I'm guessing that your mother is elderly, as well. If I was you, I wouldn't want there to be such a dark cloud over the last years of my relationship with my mother.
Consider this... If your mother is elderly and her cat is elderly... Maybe they are ushering EACH OTHER to the next life.
Taking the cat away, putting her into a shelter or euthanizing her is only turning a beautiful thing into pain and heartbreak.