To deny that “people have a moral right to make money off exploiting animals” is to confuse self-control with self-ownership.
All living organisms control themselves and their own actions to a greater or lesser extent. A cheetah decides by and for itself whom to hunt and where to rest in a limited sense. We say that it is guided by “instinct,” whatever that means. But it moves itself; no external force moves it directly. Humans exercise self-control superlatively and rationally, but for our purposes, the difference is in degree not kind.
However, it can be proven that humans are also self-owners in a legal and moral sense. No such proof is available for animals. Since an animal does not own itself (despite controlling itself), it can be legitimately owned by a second party, such as a human being.