Animal rights is a separate subject from tzar balei chaim.
Animals do not have any rights in halacha or American law. Humans have rights and responsibilities. Some human responsibilities are related to the proper treatment of animals. Halachically, this includes avoiding tzar balei chaim. But the halacha is chal on the human not on the animal.
Saying tzar balei chaim is because of animal rights is like saying shatnez is because of wool and linen rights.